Products and Intersections of Prime-Power Ideals in Leavitt Path Algebras
aa r X i v : . [ m a t h . R A ] J a n Products and Intersections of Prime-Power Ideals inLeavitt Path Algebras
Zachary Mesyan and Kulumani M. Rangaswamy
Abstract
We continue a very fruitful line of inquiry into the multiplicative ideal theory of anarbitrary Leavitt path algebra L . Specifically, we show that factorizations of an idealin L into irredundant products or intersections of finitely many prime-power idealsare unique, provided that the ideals involved are powers of distinct prime ideals. Wealso characterize the completely irreducible ideals in L , which turn out to be prime-power ideals of a special type, as well as ideals that can be factored into products orintersections of finitely many completely irreducible ideals. Keywords:
Leavitt path algebra, product of ideals, intersection of ideals, prime ideal,prime-power ideal, completely irreducible ideal
The ideal theory of the Leavitt path algebra L = L K ( E ) of a directed graph E over a field K has been an active area of research in recent years. In particular, a number of papers havebeen devoted to characterizing special types of ideals of L in terms of graphical propertiesof E , and to describing the ideals of L that can be factored into products of ideals of thesetypes. More specifically prime, primitive, semiprime, and irreducible ideals have receivedsuch treatment in the literature–see [2, 3, 4, 5, 12, 14]. An interesting feature of Leavittpath algebras is that, while they are highly noncommutative, multiplication of their idealsis commutative, and further, their ideals share a number of properties with ideals in variouscommutative rings, such as Dedekind domains (where ideals are projective), B´ezout rings(where finitely-generated ideals are principal), arithmetical rings (where the ideal lattices aredistributive), and Pr¨ufer domains (where the ideal lattices have special properties). Takentogether, the aforementioned literature has produced a rich multiplicative ideal theory forLeavitt path algebras. Our goal in this paper is to contribute to this line of inquiry byexamining ideals in Leavitt path algebras that can be factored into finite products andintersections of prime-power ideals and variants thereof.After recalling the relevant background material (Section 2), in Section 3 we show thatin any Leavitt path algebra L , factorizations of an ideal into irredundant products of prime-power ideals are unique, up to reordering the factors, provided that they are powers ofdifferent prime ideals (Theorem 3.2). A consequence of this is that factorizations of anideal of L into irredundant products of prime ideals are unique, up to reordering the factors1Corollary 3.4). Then, in Section 4, we deduce that decompositions of an ideal of L into irre-dundant intersections of finitely many prime-power ideals are likewise unique (Theorem 4.6),using the fact that the product of any finite collection of powers of distinct prime ideals in L is equal to its intersection (Proposition 4.3).In Section 5 we classify the completely irreducible ideals in an arbitrary Leavitt pathalgebra, showing that they are prime-power ideals of a special sort. Recall that a properideal I of a ring R is irreducible if I is not the intersection of any two ideals of R properlycontaining I , and I is completely irreducible if I is not the intersection of any set of idealsproperly containing I . For example, every prime ideal in any ring is clearly irreducible,but the zero ideal in Z is not completely irreducible despite being prime. Moreover, everymaximal ideal in any ring is trivially completely irreducible. In the case of commutativerings, these ideals and variants thereof were investigated in a series of papers by Fuchs,Heinzer, Mosteig, and Olberding [7, 8, 9, 10], leading to interesting factorization theorems.Irreducible ideals in Leavitt path algebras were described in [14, 3], but completely ir-reducible ideals have not received much attention before. In Theorem 5.5, we classify thecompletely irreducible ideals of an arbitrary Leavitt path algebra L . Specifically, such idealsmust either be powers of non-graded prime ideals, or be graded prime ideals of a special type.We then characterize the Leavitt path algebras L where every proper ideal is completely irre-ducible (Theorem 5.8), where every completely irreducible ideal is graded (Proposition 5.10),and where every irreducible ideal is completely irreducible (Proposition 5.11). In each casewe give equivalent conditions on L , as well as equivalent conditions on the graph E . Forexample, every proper ideal in L completely irreducible exactly when every ideal of L isgraded, and the ideals of L are well-ordered under set inclusion.In the final Section 6 we characterize the ideals in an arbitrary Leavitt path algebra thatcan be factored into products or intersections of finitely many completely irreducible ideals(Theorem 6.2), as well as the Leavitt path algebras where every proper ideal is the productor intersection of finitely many completely irreducible ideals (Theorem 6.4). Note thatevery proper ideal in any ring R is the intersection of all the completely irreducible idealscontaining it (see Proposition 5.1), and so only intersections of finitely many completelyirreducible ideals are of interest.Graphical examples are constructed illustrating many of our results. We begin with a review of the basic concepts and facts required. General Leavitt pathalgebra terminology and results can be found in [1]. For the reader familiar with thesealgebras, we suggest skipping the rest of this section and referring back as needed. A directed graph E = ( E , E , r, s ) consists of two sets E and E , where E = ∅ , togetherwith functions r, s : E → E , called range and source , respectively. The elements of E arecalled vertices , and the elements of E are called edges . We shall refer to directed graphs assimply “graphs” from now on. 2et E = ( E , E , r, s ) be a graph. A path µ in E is a finite sequence e · · · e n of edges e , . . . , e n ∈ E such that r ( e i ) = s ( e i +1 ) for i ∈ { , . . . , n − } . Here we define s ( µ ) := s ( e )to be the source of µ , r ( µ ) := r ( e n ) to be the range of µ , and | µ | := n to be the length of µ .We view the elements of E as paths of length 0, and extend s and r to E via s ( v ) = v = r ( v )for all v ∈ E . The set of all vertices on a path µ is denoted by { µ } . A path µ = e · · · e n is closed if r ( e n ) = s ( e ). A closed path µ = e · · · e n is a cycle if s ( e i ) = s ( e j ) for all i = j .An exit for a path µ = e · · · e n is an edge f ∈ E \ { e , . . . , e n } that satisfies s ( f ) = s ( e i )for some i . The graph E is said to satisfy condition (L) if every cycle in E has an exit, andto satisfy condition (K) if any vertex on a closed path µ is also the source of a closed pathdifferent from µ (i.e., one possessing a different set of edges). A cycle µ in E is said to be without (K) if no vertex along µ is the source of a different cycle in E .Given a vertex v ∈ E , we say that v is a sink if s − ( v ) = ∅ , that v is regular if s − ( v ) is finite but nonempty, and that v is an infinite emitter if s − ( v ) is infinite. A graphwithout infinite emitters is said to be row-finite . If u, v ∈ E , and there is a path µ in E satisfying s ( µ ) = u and r ( µ ) = v , then we write u ≥ v . Given a vertex v ∈ E , we set M ( v ) = { w ∈ E | w ≥ v } . A nonempty subset H of E is said to be downward directed if for any u, v ∈ H , there exists w ∈ H such that u ≥ w and v ≥ w . A subset H of E is hereditary if whenever u ∈ H and u ≥ v for some v ∈ E , then v ∈ H . Also H ⊆ E is saturated if r ( s − ( v )) ⊆ H implies that v ∈ H for any regular v ∈ E . A nonempty subset M of E is a maximal tail if it satisfies the following three conditions.(MT1) If v ∈ M and u ∈ E are such that u ≥ v , then u ∈ M .(MT2) For every regular v ∈ M there exists e ∈ E such that s ( e ) = v and r ( e ) ∈ M .(MT3) The set M is downward directed.For any H ⊆ E it is easy to see that H is hereditary if and only if M = E \ H satisfies(MT1), and H is saturated if and only if M = E \ H satisfies (MT2). Given a graph E and a field K , the Leavitt path K -algebra L K ( E ) of E is the K -algebragenerated by the set { v | v ∈ E } ∪ { e, e ∗ | e ∈ E } , subject to the following relations:(V) vw = δ v,w v for all v, w ∈ E ,(E1) s ( e ) e = er ( e ) = e for all e ∈ E ,(E2) r ( e ) e ∗ = e ∗ s ( e ) = e ∗ for all e ∈ E ,(CK1) e ∗ f = δ e,f r ( e ) for all e, f ∈ E , and(CK2) v = P e ∈ s − ( v ) ee ∗ for all regular v ∈ E .Throughout this article, K will denote an arbitrary field, E will denote an arbitrary graph,and L K ( E ) will often be denoted simply by L .For all v ∈ E we define v ∗ := v , and for all paths µ = e · · · e n ( e , . . . , e n ∈ E ) weset µ ∗ := e ∗ n · · · e ∗ , r ( µ ∗ ) := s ( µ ), and s ( µ ∗ ) := r ( µ ). It is easy to see that every elementof L K ( E ) can be expressed in the form P ni =1 a i µ i ν ∗ i for some a i ∈ K and paths µ i , ν i . Wealso note that while L K ( E ) is generally not unital, it has local units . That is, for any finitesubset { r , . . . , r n } of L K ( E ) there is an idempotent u ∈ L K ( E ) (which can be taken to bea sum of vertices in E ) such that ur i = r i = r i u for all 1 ≤ i ≤ n .3e also recall that every Leavitt path algebra L K ( E ) is Z -graded (where Z denotes thegroup of integers). Specifically, L K ( E ) = L n ∈ Z L n , where L n = (cid:26) X i a i µ i ν ∗ i ∈ L K ( E ) (cid:12)(cid:12)(cid:12) n = | µ i | − | ν i | (cid:27) . Here the homogeneous components L n are abelian subgroups satisfying L m L n ⊆ L m + n for all m, n ∈ Z . An ideal I of L K ( E ) is said to be graded if I = L n ∈ Z ( I ∩ L n ). Equivalently, I is graded if r i j ∈ I for all j ∈ { , . . . , m } , whenever r i + · · · + r i m ∈ I , with i , . . . , i m distinctand r i j ∈ L i j for each j . Next we record various results from the literature and basic observations about ideals inLeavitt path algebras that will be used frequently in the paper.Given a graph E , a breaking vertex of a hereditary saturated subset H of E is an infiniteemitter v ∈ E \ H with the property that 0 < | s − ( v ) ∩ r − ( E \ H ) | < ℵ . The set of allbreaking vertices of H is denoted by B H . For each v ∈ B H , we set v H := v − P s ( e )= v, r ( e ) / ∈ H ee ∗ .Given a hereditary saturated H ⊆ E and S ⊆ B H , we say that ( H, S ) is an admissible pair ,and the ideal of L K ( E ) generated by H ∪ { v H | v ∈ S } is denoted by I ( H, S ). The gradedideals of L K ( E ) are precisely the ideals of the form I ( H, S ) [1, Theorem 2.5.8]. Moreover,setting ( H , S ) ≤ ( H , S ) whenever H ⊆ H and S ⊆ H ∪ S , defines a partial order onthe set of all admissible pairs of L K ( E ). The map ( H, S ) I ( H, S ) gives a one-to-one order-preserving correspondence between the partially ordered set of admissible pairs and the setof all graded ideals of L K ( E ), ordered by inclusion. Finally, for any admissible pair ( H, S )we have L K ( E ) /I ( H, S ) ∼ = L K ( E \ ( H, S )), via a graded isomorphism (i.e., one that takeseach homogeneous component in one ring to the corresponding homogeneous component inthe other ring) [1, Theorem 2.4.15]. Here E \ ( H, S ) is a quotient graph of E , where( E \ ( H, S )) = ( E \ H ) ∪ { v ′ | v ∈ B H \ S } and ( E \ ( H, S )) = { e ∈ E | r ( e ) / ∈ H } ∪ { e ′ | e ∈ E with r ( e ) ∈ B H \ S } , and r, s are extended to E \ ( H, S ) by setting s ( e ′ ) = s ( e ) and r ( e ′ ) = r ( e ) ′ . (We note that( E \ ( H, B H )) = E \ H for any hereditary saturated H .) More specifically, the aforemen-tioned isomorphism L K ( E ) /I ( H, S ) ∼ = L K ( E \ ( H, S )) preserves the vertices in E \ H andedges e ∈ E whose ranges are not in H , along with the corresponding ghost edges e ∗ . Theorem 2.1.
Let L = L K ( E ) be a Leavitt path algebra, and let I be an ideal of L , with H = I ∩ E and S = { v ∈ B H | v H ∈ I } . (1) [13, Theorem 4] I = I ( H, S ) + P i ∈ Y h f i ( c i ) i where Y is a possibly empty index set;each c i is a cycle without exits in E \ ( H, S ) ; and each f i ( x ) ∈ K [ x ] is a polynomialwith a nonzero constant term, which is of smallest degree such that f i ( c i ) ∈ I . (2) [1, Proposition 2.4.7] Using the notation of (1) , we have I/I ( H, S ) = L i ∈ Y h f i ( c i ) i ⊆ L i ∈ Y h{ c i }i , where h{ c i }i is the ideal of L/I ( H, S ) generated by { c i } . [1, Corollary 2.4.16] I ( H, S ) ∩ E = H = h H i ∩ E , where h H i is the ideal of L generated by H . (4) [1, Corollary 2.8.17] For any ideal J of L , we have IJ = J I . For an ideal I of L , expressed as in Theorem 2.1(1), we refer to I ( H, S ), also denotedgr( I ), as the graded part of I . Lemma 2.2. [14, Lemma 3.1] Let I be a graded ideal of a Leavitt path algebra L . (1) IJ = I ∩ J for any ideal J of L . In particular, for any ideal J ⊆ I , we have IJ = J . (2) Let I , . . . , I n be ideals of L , for some positive integer n . Then I = I · · · I n if and onlyif I = I ∩ · · · ∩ I n . Lemma 2.3.
Let L = L K ( E ) be a Leavitt path algebra, let c and d be distinct cycles in E (that is, c and d have distinct sets of edges) without exits, and let f ( x ) , g ( x ) ∈ K [ x ] bepolynomials with nonzero constant terms. (1) [14, Lemma 3.3] h f ( c ) ih g ( c ) i = h f ( c ) g ( c ) i . (2) [3, Lemma 2.2(2)] h f ( c ) ih g ( d ) i = { } . Lemma 2.4.
Let L = L K ( E ) be a Leavitt path algebra and c a cycle without exits in E . (1) [1, Lemma 2.7.1] h{ c }i ∼ = M Y ( K [ x, x − ]) for some index set Y . (2) [12, Lemma 3.5] If E is downward directed, and I is a nonzero ideal of L containingno vertices, then I = h f ( c ) i ⊆ h{ c }i , for some f ( x ) ∈ K [ x ] with a nonzero constantterm. (3) If E is downward directed, and I is an ideal of L containing a vertex, then h{ c }i ⊆ I .Proof of (3). First note that E being downward directed, and c having no exits, means that u ≥ s ( c ) for all u ∈ E . So if an ideal I of L contains a vertex u , then I ∩ E being hereditary(see Theorem 2.1) implies that s ( c ) ∈ I . It follows that h{ c }i ⊆ I . Remark 2.5.
We note, for future reference, that in the isomorphism h{ c }i ∼ = M Y ( K [ x, x − ])constructed in the proof of [1, Lemma 2.7.1], c ∈ h{ c }i is sent to a matrix in M Y ( K [ x, x − ])which has x as one of the entries and zeros elsewhere. Theorem 2.6. [12, Theorem 3.12] Let L = L K ( E ) be a Leavitt path algebra, let I be aproper ideal if L , and let H = I ∩ E . Then I is a prime ideal if and only if I satisfies oneof the following conditions. (1) I = I ( H, B H ) , and E \ H is downward directed. (2) I = I ( H, B H \ { u } ) for some u ∈ B H , and E \ H = M ( u ) . (3) I = I ( H, B H ) + h f ( c ) i where c is a cycle without (K), E \ H = M ( s ( c )) , and f ( x ) ∈ K [ x, x − ] is an irreducible polynomial. emma 2.7. [14, Corollary 4.5] Let L be a Leavitt path algebra, let P be a prime ideal of L , and let I be an ideal of L . If P ( I , then P = IP . Remark 2.8.
An interesting consequence of Lemma 2.7 is that if P ( I , for some primeideal P and ideal I of L , then P ( I n for all positive integers n . (For, P = IP implies that P = I n P ⊆ I n , and the inclusion must be proper, since otherwise we would have P = I .) Proposition 2.9. [3, Proposition 3.2] Let L = L K ( E ) be a Leavitt path algebra, let I be aproper ideal of L , and write gr( I ) = I ( H, S ) . Then the following are equivalent. (1) I is irreducible. (2) I is a prime-power ideal. (3) Either (3 . I is a graded prime ideal, in which case ( E \ ( H, S )) is downward directed, or (3 . I is a power of a non-graded prime ideal, in which case ( E \ ( H, S )) is downwarddirected; and I = I ( H, B H ) + h p n ( c ) i for a ( unique ) cycle c without exits in E \ ( H, B H ) , an irreducible polynomial p ( x ) ∈ K [ x, x − ] , and a positive integer n . We conclude this section with a couple of facts about ideals in general rings.
Proposition 2.10. [5, Proposition 1] Let R be a ring with local units and Y a nonemptyset. (1) Every ideal of M Y ( R ) is of the form M Y ( I ) for some ideal I of R . The map I M Y ( I ) defines a lattice isomorphism between the lattice of ideals of R and the lattice of idealsof M Y ( R ) . (2) For any two ideals I and J of R , we have M Y ( IJ ) = M Y ( I ) M Y ( J ) . In this section we establish the uniqueness of factoring an arbitrary ideal in a Leavittpath algebra into an irredundant product of finitely many prime-power ideals. We beginwith a couple of technical lemmas.
Lemma 3.1.
Let L = L K ( E ) be a Leavitt path algebra, and let I = P r · · · P r m m , where each P i is a prime ideal, m, r , . . . , r m are positive integers, P i P j whenever i = j , and r i = 1 whenever P i is graded. Then the P i , the r i , and m are uniquely determined by I , up to theorder in which the ideals appear in the product.Proof. Suppose that I = Q s · · · Q s n n , where each Q i is a prime ideal, n, s , . . . , s n are positiveintegers, Q i Q j whenever i = j , and s i = 1 whenever Q i is graded. We shall prove that m = n , and that there is a permutation σ of { , . . . , m } , such that P i = Q σ ( i ) and r i = s σ ( i ) for each i ∈ { , . . . , m } . 6et i ∈ { , . . . , m } . Since P i is prime, I ⊆ P i implies that Q σ ( i ) ⊆ P i for some σ ( i ) ∈{ , . . . , n } . Likewise, since I ⊆ Q σ ( i ) , we have P i ′ ⊆ Q σ ( i ) for some i ′ ∈ { , . . . , m } , andhence P i ′ ⊆ P i . By hypothesis, this means that i ′ = i , and hence P i = Q σ ( i ) . Since i wasarbitrary, and since P , . . . , P m are distinct, this implies that m ≤ n . Then, by symmetry,we conclude that m = n , and that i σ ( i ) defines a permutation σ of { , . . . , m } . Setting t i = s σ ( i ) for each i , and using Theorem 2.1(4), we then have I = P r P r · · · P r m m = P t P t · · · P t m m . So it remains to show that r i = t i for each i .By hypothesis, if P i is graded, for some i , then r i = 1 = t i . We may therefore assumethat at least one of the ideals P , . . . , P m is non-graded, which may be taken to be P , byTheorem 2.1(4). (Note that this implies that none of P , . . . , P m is zero.) We shall concludethe proof by showing that r = t .By Theorem 2.6, P = I ( H, B H ) + h f ( c ) i , where H = P ∩ E , c is a cycle withoutexits in E \ ( H, B H ), E \ H = M ( s ( c )), and f ( x ) ∈ K [ x, x − ] is irreducible (and may betaken to be in K [ x ], with a nonzero constant term). Note that ( E \ ( H, B H )) = E \ H is downward directed. We shall now pass to the ring ¯ L = L/I ( H, B H ), which may beidentified with L K ( E \ ( H, B H )), as mentioned in Section 2.3. For each i ∈ { , . . . , m } let¯ P i = ( P i + I ( H, B H )) /I ( H, B H ) ⊆ ¯ L , and let M be the ideal of ¯ L generated by { c } . Then,by Theorem 2.1, ¯ P = h f ( c ) i ( M . Moreover, by Lemma 2.4(2,3), for each i ≥
2, either M ⊆ ¯ P i or ¯ P i ⊆ M , depending on whether or not ¯ P i contains a vertex.Without loss of generality, we may assume that ¯ P i ( M if 1 < i ≤ k , and M ⊆ ¯ P i if k < i ≤ m , for some k . Then, by Lemma 2.2(1), M being graded implies that ¯ P i M =¯ P i ∩ M = ¯ P i for each 1 < i ≤ k , and M ¯ P i = M ∩ ¯ P i = M for each k < i ≤ m . Using theseequations repeatedly, we then have¯ P r ¯ P r · · · ¯ P r m m = ¯ P r ¯ P r · · · ¯ P r k k M ¯ P r k +1 k +1 · · · ¯ P r m m = ¯ P r ¯ P r · · · ¯ P r k k M = ¯ P r ¯ P r · · · ¯ P r k k , and likewise ¯ P t ¯ P t · · · ¯ P t m m = ¯ P t ¯ P t · · · ¯ P t k k . Now, for each 1 < i ≤ k , since ¯ P i contains no vertices, we have ¯ P i = h g i ( c ) i for some g i ( x ) ∈ K [ x ] with a nonzero constant term, by Lemma 2.4(2). This means that, for each1 < i ≤ k , we have gr( P i ) ⊆ I ( H, B H ), which in turn implies that P i is not graded.(Otherwise, P i = gr( P i ) ⊆ I ( H, B H ) ⊆ P , contrary to hypothesis.) Thus, by Theorem 2.6,for each 1 < i ≤ k , we have P i = I ( H i , B H i ) + h f i ( c i ) i where H i = P i ∩ E , c i is a cyclewithout exits in E \ ( H i , B H i ), E \ H i = M ( s ( c i )), and f i ( x ) ∈ K [ x, x − ] is irreducible (andmay be taken to be in K [ x ], with a nonzero constant term). In particular, H i ⊆ H .We claim that H i = H for each 1 < i ≤ k . Seeking a contradiction, suppose that thereis a vertex u ∈ H \ H i for some i . Since E \ H i = M ( s ( c i )), we have u ≥ s ( c i ). Since H is hereditary, this implies that s ( c i ) ∈ H , and hence { c i } ⊆ H . It follows that P i ⊆ P ,contrary to hypothesis. Thus H i = H , and hence also E \ ( H, B H ) = E \ ( H i , B H i ), for each1 < i ≤ k .Since E \ H = M ( s ( c )), and the cycles c i have no exits in E \ ( H, B H ), necessarily c i = c for each 1 < i ≤ k . Thus P i = I ( H, B H ) + h f i ( c ) i for each 1 < i ≤ k , and7he polynomials f i ( x ) are necessarily non-conjugate (i.e., are not scalar multiples of oneanother). By Lemma 2.3(1), we therefore have h f r ( c ) i · · · h f r k k ( c ) i = ¯ P r ¯ P r · · · ¯ P r k k = ¯ I = ¯ P t ¯ P t · · · ¯ P t k k = h f t ( c ) i · · · h f t k k ( c ) i . Setting v = s ( c ), and noting that vf i ( c ) v = f i ( c ) for each i , we see that ¯ I ⊆ v ¯ Lv . Now,according to [1, Lemma 2.2.7], given that c is a cycle without exits in E \ ( H, B H ), we have vL K ( E \ ( H, B H )) v ∼ = K [ x, x − ], via the K -algebra isomorphism that maps v c x ,and c ∗ x − . In particular, f i ( c ) is mapped to f i ( x ) by this isomorphism, and so h f r ( x ) · · · f r k k ( x ) i = h f r ( x ) i · · · h f r k k ( x ) i = h f t ( x ) i · · · h f t k k ( x ) i = h f t ( x ) · · · f t k k ( x ) i in K [ x, x − ]. This implies that f r ( x ) divides f t ( x ) · · · f t k k ( x ), and that f t ( x ) divides f r ( x ) · · · f r k k ( x ). Since K [ x, x − ] is a unique factorization domain, and the f i ( x ) are non-conjugate and irreducible, we then conclude that r = t . By symmetry, it follows that r i = t i for all i .Recall that a (finite) product I = J · · · J m of ideals J i in a ring is irredundant if either m = 1 or I = J · · · J i − J i +1 · · · J m for all i ∈ { , . . . , m } .We are now ready for our first main result. Theorem 3.2.
Let L be a Leavitt path algebra, and let I , . . . , I m be powers of distinct primeideals of L . If I · · · I m is irredundant, then the prime-power ideals I , . . . , I m are uniquelydetermined, up to the order in which they appear.Proof. For each i , write I i = P r i i for some prime ideal P i and positive integer r i , where wecan let r i = 1 if P i is graded, by Lemma 2.2(1). By Lemma 3.1, to conclude that the I i areuniquely determined, up to the order in which they appear, it suffices to show that P i P j whenever i = j (assuming that m > P i ⊆ P j for some i = j . Since these ideals aredistinct, by hypothesis, we have P i ( P j . Hence, by Lemma 2.7, P j P i = P i , and so P r j j P r i i = P r i i . By Theorem 2.1(4), we then have I · · · I m = I · · · I j − I j +1 · · · I m , contrary to hypothesis. Therefore P i P j whenever i = j , as desired. Remark 3.3.
The assumption that the I i are powers of distinct prime ideals is necessaryfor the conclusion of Theorem 3.2 to hold. For example, if P is a non-graded prime ideal ina Leavitt path algebra L , then P = P , by Lemma 3.1. Thus P and P · P are two distinctirredundant representations of the same ideal as a product of prime-power ideals.The following consequence of Theorem 3.2 significantly strengthens [4, Theorem 3.16]and [6, Theorem 3.3]. Corollary 3.4.
Let L be a Leavitt path algebra, and let I be an irredundant product of finitelymany prime ideals of L . Then the prime ideals in the product I are uniquely determined, upto the order in which they appear.Proof. By Theorem 2.1(4) and the hypothesis, we can write I = J · · · J m , where J , . . . , J m are powers of distinct prime ideals, and the product is irredundant. The desired conclusionnow follows from Theorem 3.2. 8 Intersections of Prime-Power Ideals
Our next goal is to show that for powers of distinct prime ideals in a Leavitt pathalgebra, the product coincides with the intersection. This result will be useful for translatingstatements about products of ideals into statements about intersections. We require twotechnical lemmas.
Lemma 4.1.
Let L = L K ( E ) be a Leavitt path algebra, let ( H, S ) be an admissible pair, let c be a cycle without exits in E \ ( H, S ) , and let m be a positive integer. For each i ∈ { , . . . , m } let I i = I ( H, S ) + h f i ( c ) i , where f i ( x ) ∈ K [ x, x − ] is nonzero. If gcd( f i ( x ) , f j ( x )) = 1 forall i = j , then I · · · I m = I ( H, S ) + h f ( c ) · · · f m ( c ) i = I ∩ · · · ∩ I m . Proof.
For each i write ¯ I i = I i /I ( H, S ). Then, in L K ( E \ ( H, S )) ∼ = L K ( E ) /I ( H, S ), we have¯ I i = h f i ( c ) i ⊆ h{ c }i . Thus, in this ring, using Lemma 2.3(1), we have¯ I · · · ¯ I m = h f ( c ) i · · · h f m ( c ) i = h f ( c ) · · · f m ( c ) i . Now, the hypothesis that gcd( f i ( x ) , f j ( x )) = 1 for all i = j implies that in K [ x, x − ] we havelcm( f ( x ) , . . . , f m ( x )) = f ( x ) · · · f m ( x ), and so h f ( x ) · · · f m ( x ) i = h lcm( f ( x ) , . . . , f m ( x )) i = h f ( x ) i ∩ · · · ∩ h f m ( x ) i . By Lemma 2.4(1), Remark 2.5, and Proposition 2.10, we then have¯ I · · · ¯ I m = h f ( c ) · · · f m ( c ) i = h f ( c ) i ∩ · · · ∩ h f m ( c ) i = ¯ I ∩ · · · ∩ ¯ I m . It follows that I · · · I m = I ∩ · · · ∩ I m and I · · · I m = I ( H, S ) + h f ( c ) · · · f m ( c ) i , as desired. Lemma 4.2.
Let L = L K ( E ) be a Leavitt path algebra, let m be a positive integer, and let H , . . . , H m be distinct hereditary saturated subsets of E . Also, for each i ∈ { , . . . , m } ,let c i be a cycle without exits in E \ ( H i , B H i ) such that E \ H i = M ( s ( c i )) , and let I i = I ( H i , B H i ) + h f i ( c i ) i for some f i ( x ) ∈ K [ x, x − ] . Then I · · · I m = I ∩ · · · ∩ I m .Proof. Since the c i are without exits in E \ ( H i , B H i ), the H i are distinct, and E \ H i = M ( s ( c i )) for each i , necessarily the cycles c i are distinct. Letting N = I ∩ · · · ∩ I m , byTheorem 2.1(1), N = I ( H, S ) + P i ∈ Y h g i ( d i ) i , where each d i is a cycle without exits in E \ ( H, S ), and each g i ( x ) ∈ K [ x ] has a nonzero constant term. By Lemma 2.2(2), I ( H, S ) = I ( H , B H ) ∩ · · · ∩ I ( H m , B H m ) = I ( H , B H ) · · · I ( H m , B H m ) ⊆ I · · · I m . Since I · · · I m ⊆ I ∩ · · · ∩ I m for any ideals I i in any ring, it therefore suffices to show that h g i ( d i ) i ⊆ I · · · I m for arbitrary i ∈ Y .It cannot be the case that d i ∈ I ( H , B H ) ∩ · · · ∩ I ( H m , B H m ), since then we would have d i ∈ I ( H, S ), contrary to the choice of d i . Thus, without loss of generality we may assumethat d i / ∈ I ( H , B H ). Then d i is a cycle in E \ ( H , B H ). Since E \ ( H , B H ) is a subgraphof E \ ( H, S ), we conclude that d i has no exits in E \ ( H , B H ). The hypotheses on c then9mply that d i = c . Thus it cannot be the case that d i / ∈ I ( H j , B H j ) for some j >
1, sinceotherwise the same argument would show that d i = c j , contradicting the distinctness of thecycles c j . Therefore, by Lemma 2.2(1), we have g i ( d i ) ∈ I ∩ m \ j =2 I ( H j , B H j ) = I · m Y j =2 I ( H j , B H j ) ⊆ I · · · I m , which implies that h g i ( d i ) i ⊆ I · · · I m , as desired.We are now ready to show that for powers of distinct prime ideals in a Leavitt pathalgebra, the product coincides with the intersection. A version of this result, for row-finitegraphs, can be found in an unpublished thesis of van den Hove [11, Corollary 5.3.3], wherethe proof uses lattice-theoretic properties of ideals, represented as ordered triples of vertices,cycles, and polynomials. Proposition 4.3.
Let L = L K ( E ) be a Leavitt path algebra, let m, r , . . . , r m be positiveintegers, and let P , . . . , P m be distinct prime ideals of L . Then P r · · · P r m m = P r ∩ · · · ∩ P r m m . Proof.
Upon reindexing, we may assume that P k +1 , . . . , P m are graded, and P , . . . , P k arenot, for some 0 ≤ k ≤ m . Then, by Lemma 2.2(1),( P r ∩ · · · ∩ P r k k ) P r k +1 · · · P r m m = P r ∩ · · · ∩ P r m m , and so, by Theorem 2.1(4), it suffices to show that P r · · · P r k k = P r ∩ · · · ∩ P r k k , where we may assume that k ≥ P ) = · · · = gr( P n ) = I ( H, B H ) for some1 < n ≤ k . By Theorem 2.6, there can be only one cycle c with no exits in E \ ( H, B H ), andso P i = I ( H, B H ) + h p i ( c ) i , where p i ( x ) ∈ K [ x, x − ] is irreducible, for each 1 ≤ i ≤ n . Sincethe P i are distinct, necessarily gcd( p i ( x ) , p j ( x )) = 1, and hence also gcd( p i ( x ) r i , p j ( x ) r j ) = 1,for all i = j . Therefore, applying Lemma 4.1 to the P r i i = I ( H, B H ) + h p i ( x ) r i i , we concludethat P r · · · P r n n = P r ∩ · · · ∩ P r n n = I ( H, B H ) + h p ( c ) r · · · p n ( c ) r n i . In this fashion, using Theorem 2.1(4), we can group the ideals P r i i in P r · · · P r k k having thesame graded part, to obtain a product of ideals of the from I ( H, B H ) + h f ( c ) i , which satisfythe hypotheses of Lemma 4.2. The desired conclusion then follows from that lemma. Remark 4.4.
The previous result implies that an ideal in a Leavitt path algebra is theproduct of prime-power ideals if and only if it is the intersection of finitely many prime-power ideals. In contrast to this, a product of prime ideals may not be expressible as anintersection of prime ideals. For example, if P is a non-graded prime ideal in a Leavitt pathalgebra, then P is not the intersection of any collection of prime ideals, according to [3,Lemma 4.9]. (This can also be concluded from Theorem 4.6.)10 emark 4.5. We note also that Proposition 4.3 describes a property that is special toLeavitt path algebras, which does not in general hold even for commutative unital rings.For example, let K be a field, let R = K [ x, y ] be the polynomial ring in two variables, andconsider the ideals I = h x, y i and J = h x i of R . Then it is easy to see that I and J areprime (and distinct), but IJ = h x , xy i 6 = J = I ∩ J .Recall that given a nonempty collection { S i | i ∈ Y } of sets, the intersection T i ∈ Y S i is irredundant if either | Y | = 1, or T i ∈ Y \{ j } S i S j for all j ∈ Y . Similarly, the union S i ∈ Y S i is irredundant if either | Y | = 1, or S j S i ∈ Y \{ j } S i for all j ∈ Y .We are now ready for the main result of this section, which is an analogue of Theorem 3.2for intersections. Theorem 4.6.
Let L be a Leavitt path algebra, and let I be an irredundant intersection offinitely many prime-power ideals of L . Then the ideals in the intersection I are uniquelydetermined.Proof. Suppose that P r ∩ · · · ∩ P r m m = Q s ∩ · · · ∩ Q s n n are irredundant intersections, where m, n, r i , s i are positive integers, and the P i and Q i areprime ideals of L . By Lemma 2.2(1), P r i i = P i whenever P i is graded, and likewise for the Q i . So we may assume that r i = 1 whenever P i is graded, and likewise for the Q i . We mayalso assume that m >
1, since otherwise there is nothing to prove.Next, suppose that P i ⊆ P j for some i = j . Since the intersection is irredundant, the P i must be distinct, and so P i ( P j . Then, by Lemma 2.7, P j P i = P i . It follows that P r j j P r i i = P r i i , which in turn implies that P r j j ∩ P r i i = P r i i . We can then replace P r j j ∩ P r i i with P r i i in the intersection P r ∩ · · · ∩ P r m m , which contradicts the supposition that it isirredundant. Thus P i P j for all i = j , and likewise for the Q i .Since the P i are distinct, as are the Q i , by Proposition 4.3, we have P r · · · P r m m = Q s · · · Q s n n . Then Lemma 3.1 implies that m = n , and that there is a permutation σ of { , . . . , m } , suchthat P i = Q σ ( i ) and r i = s σ ( i ) for each i ∈ { , . . . , m } , giving the desired conclusion.We conclude this section with a couple of technical lemmas, which will useful subse-quently. The first is an expanded version of part of the proof of [14, Theorem 6.2]. Lemma 4.7.
Let L = L K ( E ) be a Leavitt path algebra, let I be a non-graded ideal of L , andwrite I = I ( H, S ) + P i ∈ Y h f i ( c i ) i , using the notation of Theorem 2.1(1). Suppose furtherthat I ( H, S ) = T mi =1 P i for some graded prime ideals P i = I ( H i , S i ) . Then the following hold. (1) For each i ∈ Y there is an l ∈ { , . . . , m } such that s ( c i ) / ∈ P l . Moreover, s ( c j ) ∈ P l for all j = i , S l = B H l , and u ≥ s ( c i ) for every u ∈ E \ H l . (2) | Y | ≤ m . (3) Given i ∈ Y , if l, k ∈ { , . . . , m } are such that s ( c i ) / ∈ P l ∪ P k , then P l ∩ P k is a gradedprime ideal. oreover, if T mi =1 P i is an irredundant intersection, then the following hold. (4) Upon reindexing the c i and P i , and letting | Y | = n ≤ m , for each i ∈ { , . . . , n } wehave s ( c i ) / ∈ P i and s ( c i ) ∈ P j for all j ∈ { , . . . , m } \ { i } . (5) I = Q ni =1 ( I ( H i , B H i ) + h f i ( c i ) i ) · Q mi = n +1 I ( H i , B H i ) , where the P i = I ( H i , B H i ) areindexed as in (4) .Proof. (1) Let i ∈ Y . Then there is an l ∈ { , . . . , m } such that s ( c i ) / ∈ P l , since otherwisewe would have s ( c i ) ∈ T mi =1 P i , and hence c i ∈ I ( H, S ), contrary to hypothesis.Since c i is a cycle without exists in E \ ( H, S ), and (
H, S ) ≤ ( H l , S l ), it follows that c i hasno exits in E \ ( H l , S l ) as well. Now, by Theorem 2.6, E \ H l is downward directed, and so wemust have u ≥ s ( c i ) for every u ∈ E \ H l . Then it cannot be the case that s ( c i ) ∈ B H l and S l = B H l \{ s ( c i ) } , since in that situation c i would have an exit in E \ ( H l , S l ), by construction.Thus, by Theorem 2.6, S l = B H l . Moreover, since E \ H l is downward directed, it cannotbe the case that s ( c j ) ∈ E \ H l for some cycle c j (without exists) that is different from c i .Hence s ( c j ) ∈ H l ⊆ P l for all j = i .(2) This follows immediately from (1).(3) Suppose that s ( c i ) / ∈ P l ∪ P k for some l, k ∈ { , . . . , m } , and let P ′ = P l ∩ P k . Then,by (1), u ≥ s ( c i ) for every u ∈ ( E \ H l ) ∪ ( E \ H k ), S l = B H l , and S k = B H k . Being anintersection of graded ideals, P ′ is graded, and so P ′ = I ( H ′ , S ′ ) for some admissible pair( H ′ , S ′ ). Thus, by Theorem 2.1(3), we have H ′ = P ′ ∩ E = ( P l ∩ E ) ∩ ( P k ∩ E ) = H l ∩ H k , and hence ( E \ H l ) ∪ ( E \ H k ) = E \ ( H l ∩ H k ) = E \ H ′ . Since u ≥ s ( c i ) for every u ∈ ( E \ H l ) ∪ ( E \ H k ) = E \ H ′ , it follows that E \ H ′ is downwarddirected. Thus to conclude that P ′ is a (graded) prime ideal, it suffices to show that S ′ = B H ′ ,by Theorem 2.6.Consider u ∈ B H ′ . Since u ∈ E \ H ′ , we have u ≥ s ( c i ), from which it follows that u ∈ B H l ∩ B H k . Let e , . . . , e n l , f , . . . , f n k , g , . . . , g n ′ ∈ E be all the edges having source u but range in E \ H ′ , where each r ( e i ) ∈ H l \ H k , r ( f i ) ∈ H k \ H l , and r ( g i ) / ∈ H l ∪ H k . Thenfor each i we have e i e ∗ i ∈ P l and f i f ∗ i ∈ P k , and so u H ′ = u − n l X i =1 e i e ∗ i − n k X i =1 f i f ∗ i − n ′ X i =1 g i g ∗ i = u H l − n l X i =1 e i e ∗ i = u H k − n k X i =1 f i f ∗ i ∈ P l ∩ P k = P ′ . Therefore, u H ′ ∈ P ′ for all u ∈ B H ′ , and so S ′ = B H ′ , as desired.For the remainder of the proof we shall assume that T mi =1 P i is an irredundant intersection.(4) Suppose that s ( c i ) / ∈ P l and s ( c i ) / ∈ P k , for some i ∈ Y and distinct l, k ∈ { , . . . , m } .Then, by (3), P ′ = P l ∩ P k is a graded prime ideal. Hence, by Lemma 2.2(2), P ′ = P l P k ,and so either P l ⊆ P ′ or P k ⊆ P ′ , which means that either P l ⊆ P k or P k ⊆ P l . Thiscontradicts the hypothesis that T mi =1 P i is irredundant, and hence s ( c i ) / ∈ P j for at most one j ∈ { , . . . , m } , from which (4) follows, by (1).125) For each i ∈ { , . . . , n } let Q i = I ( H i , B H i ) + h f i ( c i ) i , and let J = T mj = n +1 P j . (If n = m , then we take J = L .) Then, by (4), we have s ( c i ) ∈ J for all i ∈ { , . . . , n } , andhence I ⊆ J . We now proceed by induction on n .If n = 1, then by Lemma 2.2(1) and the fact that ideal multiplication distributes overideal addition in any ring, Q P · · · P m = Q J = ( P + h f ( c ) i ) J = P J + h f ( c ) i J = m \ i =1 P i + h f ( c ) i J. Since I ⊆ J , by Lemma 2.2(1), we see that h f ( c ) i J = h f ( c ) i , and hence Q P · · · P m = I ( H, S ) + h f ( c ) i = I. So let us suppose that n >
1, and the statement holds for n −
1. That is, Q · · · Q n − P n +1 · · · P m = J ∩ n − \ i =1 P i + n − X i =1 h f i ( c i ) i . Then, using Theorem 2.1(4) and Lemma 2.2(1), we have Q · · · Q n P n +1 · · · P m = Q · · · Q n − ( P n + h f n ( c n ) i ) P n +1 · · · P m = Q · · · Q n − P n P n +1 · · · P m + Q · · · Q n − h f n ( c n ) i J = (cid:18) J ∩ n − \ i =1 P i + n − X i =1 h f i ( c i ) i (cid:19) P n + Q · · · Q n − h f n ( c n ) i = m \ i =1 P i + n − X i =1 h f i ( c i ) i P n + (cid:18) n − Y i =1 ( P i + h f i ( c i ) i ) (cid:19) h f n ( c n ) i . Since s ( c i ) ∈ P j for all i = j , again using Lemma 2.2(1), we see that Q · · · Q n P n +1 · · · P m = I ( H, S ) + n − X i =1 h f i ( c i ) i + (cid:18) n − Y i =1 P i (cid:19) h f n ( c n ) i + (cid:18) n − Y i =1 h f i ( c i ) i (cid:19) h f n ( c n ) i = I ( H, S ) + n X i =1 h f i ( c i ) i + (cid:18) n − Y i =1 h f i ( c i ) i (cid:19) h f n ( c n ) i Finally, h f i ( c i ) ih f j ( c j ) i ⊆ I ( H, S ) for i = j , by Lemma 2.3(2) and Theorem 2.1(2). Thus Q · · · Q n P n +1 · · · P m = I ( H, S ) + n X i =1 h f i ( c i i = I, as desired. Lemma 4.8.
Let L = L K ( E ) be a Leavitt path algebra, and let I be a non-graded ideal of L such that gr( I ) is a prime ideal. Then there exist non-graded prime ideals P , . . . , P m andpositive integers m, r , . . . , r m such that I = P r · · · P r m m = P r ∩ · · · ∩ P r m m , and gr( P i ) = gr( I ) for each i . roof. By Lemma 4.7(2,5), we have I = I ( H, B H ) + h f ( c ) i , where gr( I ) = I ( H, B H ) is aprime ideal, c is a cycle without exits in E \ ( H, B H ), and f ( x ) ∈ K [ x ] is a polynomial witha nonzero constant term.Write f ( x ) = p r ( x ) · · · p r m m ( x ) for some non-conjugate irreducible polynomials p i ( x ) ∈ K [ x ] and positive integers m, r , . . . , r m . Then, using Lemma 2.3(1) and Theorem 2.1(2), wehave I = ( I ( H, B H ) + h p ( c ) i ) r · · · ( I ( H, B H ) + h p m ( c ) i ) r m . Writing P i = I ( H, B H ) + h p i ( c ) i for each i , we then have I = P r · · · P r m m , where each P i isprime, by Theorem 2.6. By construction, the P i are distinct, and so I = P r ∩ · · · ∩ P r m m , byProposition 4.3. Recall that a proper ideal I of a ring R is completely irreducible if I is not the intersectionof any set of ideals properly containing I . In this section we characterize the completelyirreducible ideals of an arbitrary Leavitt path algebra. These ideals turn out to be prime-power ideals of a special sort, and so the results from the previous two sections apply tothem.It is well-known and easy to prove that an ideal I of a ring R is completely irreducibleif and only if there exists r ∈ R such that I is an ideal maximal with respect to r / ∈ I .An straightforward consequence of this statement is the following observation, which will beuseful throughout the rest of the paper. Proposition 5.1.
Every proper ideal in a ring is the intersection of the completely irreducibleideals that contain it.
The following concepts will help us describe the completely irreducible ideals in Leavittpath algebras.
Definition 5.2.
Let E be a graph, and let S be a nonempty subset of E .(1) We say that S satisfies the countable separation property ( CSP for short) if there is acountable subset T of S , such that for every u ∈ S there is a v ∈ T satisfying u ≥ v .(2) We say that S satisfies the strong CSP if S satisfies the CSP with respect to somecountable subset T , such that T is contained in every nonempty hereditary saturatedsubset of S .We next describe the Leavitt path algebras for which the zero ideal is completely irre-ducible. Proposition 5.3.
The following are equivalent for any Leavitt path algebra L = L K ( E ) . (1) The zero ideal is completely irreducible. (2)
The graph E satisfies condition (L), and E is downward directed and satisfies thestrong CSP. roof. (1) ⇒ (2) Suppose that { } is completely irreducible. According to [1, Proposition2.3.2], { } is the Jacobson radical of L , and hence { } is also the intersection of all theprimitive ideals of L . (It is a standard fact that in any ring, the Jacobson radical is theintersection of all the primitive ideals. This is typically proved for unital rings, but can beeasily extended to rings with local units.) We therefore conclude that { } must be primitive,and hence L is a primitive ring. By [2, Theorem 5.7], L being primitive implies that E satisfies condition (L), E is downward directed, and E satisfies the CSP, with respect toa countable nonempty subset S of E . Let { H i | i ∈ Y } be the collection of all nonemptyhereditary saturated subsets of E , and let H = T i ∈ Y H i . We shall show that E satisfiesthe strong CSP with respect to S ∩ H .First, we claim that H = ∅ . If on the contrary, H = ∅ , then, by Theorem 2.1(3), E ∩ \ i ∈ Y h H i i = \ i ∈ Y H i = H = ∅ . As the intersection of graded ideals, T i ∈ Y h H i i is itself graded, and hence E ∩ T i ∈ Y h H i i = ∅ implies that T i ∈ Y h H i i = { } , again, by Theorem 2.1(3). But this contradicts { } beingcompletely irreducible, and hence H = ∅ .To conclude the proof, let us take an arbitrary vertex v ∈ E and show that v ≥ w forsome w ∈ S ∩ H . Letting u ∈ H be any vertex, E satisfying the CSP with respect to S implies that there exist v ′ , u ′ ∈ S such that v ≥ v ′ and u ≥ u ′ . Since E is downwarddirected, there exists w ′ ∈ E such that u ′ ≥ w ′ and v ′ ≥ w ′ . Then, invoking the CSP again, w ′ ≥ w for some w ∈ S . Since H is hereditary, u ≥ w implies that w ∈ S ∩ H , as desired.(2) ⇒ (1) Suppose that E satisfies condition (L), E is downward directed, and E satisfies the strong CSP with respect to some nonempty countable S ⊆ E . According to[1, Proposition 2.2.14], E satisfying condition (L) implies that every nonzero ideal I of L contains a vertex, and hence I ∩ E = ∅ . By [1, Lemma 2.4.3], the set I ∩ E is hereditaryand saturated, and hence, by hypothesis, S ⊆ I ∩ E , for every nonzero ideal I . Thus theintersection of all the nonzero ideals of L contains S , and hence is nonzero. This shows that { } is completely irreducible. Example 5.4.
Let E be the following graph. • v / / • v / / • v Then clearly E satisfies condition (L), E is downward directed, and E satisfies the strongCSP with respect to { v } . Thus, the ideal { } is completely irreducible in L K ( E ), by theprevious proposition. (cid:3) We now utilize Proposition 5.3 to prove the main result of this section, which describes allthe completely irreducible ideals in an arbitrary Leavitt path algebra. (See Proposition 2.9for a description of the irreducible ideals.)
Theorem 5.5.
Let L = L K ( E ) be a Leavitt path algebra, and let I be a proper ideal of L .Then I is completely irreducible if and only if exactly one of the following conditions holds. (1) I = I ( H, S ) is a graded ideal, E \ ( H, S ) satisfies condition (L), and ( E \ ( H, S )) isdownward directed and satisfies the strong CSP. (In this case I is prime.) I = P n for some non-graded prime ideal P and positive integer n .Proof. Suppose that I is completely irreducible. Then, in particular, I is irreducible, and so,by Proposition 2.9, I = P n for some prime ideal P and positive integer n . If P is non-graded,then condition (2) holds. Let us therefore suppose that P is graded. Then I = P n = P ,by Lemma 2.2(1), and hence P = I = I ( H, S ), for some admissible pair (
H, S ). Since I is completely irreducible, so is the zero ideal of L K ( E \ ( H, S )) ∼ = L/I . Hence condition (1)holds, by Proposition 5.3. Note that, by Theorem 2.1(1,2) and Lemma 2.3, a power of anon-graded ideal is non-graded, and so conditions (1) and (2) are mutually exclusive.For the converse, first suppose that condition (1) holds. Then, by Proposition 5.3, thezero ideal of L K ( E \ ( H, S )) ∼ = L/I is completely irreducible, and hence so is I .Now let us suppose that condition (2) holds. Then, by Proposition 2.9, I = P n = I ( H, B H ) + h p n ( c ) i , where c is a cycle without exits in E \ ( H, B H ), p ( x ) ∈ K [ x, x − ] isirreducible, and ( E \ ( H, B H )) is downward directed. We shall show that ¯ P n = P n /I ( H, B H )is completely irreducible in ¯ L = L/I ( H, B H ) ∼ = L K ( E \ ( H, B H )), from which it follows that I is completely irreducible in L .First, we note that in the principal ideal domain K [ x, x − ], if h f ( x ) i ⊇ h p n ( x ) i for some f ( x ) ∈ K [ x, x − ], then p n ( x ) = f ( x ) g ( x ) for some g ( x ) ∈ K [ x, x − ], which implies that h f ( x ) i = h p k ( x ) i for some 0 ≤ k ≤ n , since p ( x ) is irreducible. Thus, the set of properideals of K [ x, x − ] containing h p n ( x ) i is precisely {h p k ( x ) i | ≤ k ≤ n } . Now let M be theideal of ¯ L generated by { c } . Then ¯ P ⊆ M , and M ∼ = M Y ( K [ x, x − ]) for some index set Y ,by Lemma 2.4(1). So, by Proposition 2.10 and Remark 2.5, the set of proper ideals of M containing ¯ P n = h p n ( c ) i is {h p k ( c ) i | ≤ k ≤ n } .Next, let J be an ideal of ¯ L such that ¯ P n ( J . Then, by Lemma 2.4(2,3), either M ⊆ J or J = h h ( c ) i ⊆ M for some h ( x ) ∈ K [ x ]. Thus, by the previous paragraph, either M ⊆ J or J = h p k ( c ) i for some k ∈ { , . . . , n − } . It follows that the intersection of all the idealsof ¯ L properly containing ¯ P n is h p n − ( c ) i . This shows that ¯ P n is completely irreducible in ¯ L ,as desired. Remark 5.6.
We note that the prime ideals of type (2) in Theorem 2.6 are always completelyirreducible. More specifically, these prime ideals are of the form I ( H, B H \{ u } ) for some u ∈ B H , where E \ H = M ( u ). In this situation ( E \ ( H, B H \{ u } )) = ( E \ H ) ∪ { u ′ } ,where u ′ is a sink. Thus the conditions in (1) of Theorem 5.5 are clearly satisfied. Remark 5.7.
Theorems 3.2 and 5.5 imply that in an irredundant product of completelyirreducible ideals, those ideals are uniquely determined, provided that they are powers ofdistinct prime ideals.Likewise, Theorems 4.6 and 5.5 imply that in an irredundant intersection of finitelymany completely irreducible ideals, those ideals are uniquely determined. However, generallyspeaking, representations of an ideal in a ring R as an irredundant intersection of an infinitecollection of completely irreducible ideals are not unique–see, e.g., [8, Example 3.1].The rest of this section is devoted to describing Leavitt path algebras where all properideals satisfy some condition related to being completely irreducible. We begin with algebraswhere all proper ideals are completely irreducible. Theorem 5.8.
The following are equivalent for any Leavitt path algebra L = L K ( E ) . Every proper ideal of L is completely irreducible. (2) Every ideal of L is graded, and the ideals of L are well-ordered under set inclusion. (3) The graph E satisfies condition (K), the admissible pairs ( H, S ) form a chain underthe partial order of admissible pairs, and ( E \ ( H, S )) satisfies the strong CSP for eachadmissible pair ( H, S ) with H = E .Proof. (1) ⇒ (2) Suppose that every proper ideal of L is completely irreducible. Then, byProposition 2.9, every proper ideal of L is a power of some prime ideal. Hence, accordingto [3, Theorem 4.1], every ideal of L is graded, and the ideals form a chain under set inclusion.Now let Y be a nonempty set of ideals of L , and let J = T I ∈ Y I . If J / ∈ Y , then J must bethe intersection of all the ideals properly containing J , and hence not completely irreducible,contrary to hypothesis. Therefore J ∈ Y , showing that Y has a least element under setinclusion. It follows that the ideals of L are well-ordered.(2) ⇒ (1) Let us take a proper ideal J of L , and show that it is completely irreducible,assuming (2). If J is maximal, then this is trivially the case, and so we may assume that J is not maximal. Let Y be the set of all ideals of L that properly contain J . Then, byhypothesis, T I ∈ Y I ∈ Y , and hence J ( T I ∈ Y I . It follows that J is completely irreducible.(1) ⇒ (3) Suppose that (1) holds. Then, by Proposition 2.9, every proper ideal of L is a power of some prime ideal. Hence, by [3, Theorem 4.1], every ideal of L is graded, E satisfies condition (K), and the admissible pairs ( H, S ) form a chain under the partialorder of admissible pairs. Given that every proper ideal of L is completely irreducible andgraded, applying Theorem 5.5, we conclude that ( E \ ( H, S )) satisfies the strong CSP foreach admissible pair ( H, S ) with H = E .(3) ⇒ (1) Assuming that (3) holds, again, by [3, Theorem 4.1], every proper ideal of L is graded and prime. Let I be a proper ideal of L , and write I = I ( H, S ), where (
H, S ) isan admissible pair. Then, by Proposition 2.9, ( E \ ( H, S )) is downward directed. Since, by(3), ( E \ ( H, S )) satisfies the strong CSP and E \ ( H, S ) satisfies condition (L), we conclude,by Theorem 5.5, that I is completely irreducible, proving (1). Example 5.9.
Let E be the following graph. • v / / (cid:7) (cid:7) Y Y • v / / (cid:7) (cid:7) Y Y • v (cid:7) (cid:7) Y Y Then E is clearly row-finite and satisfies condition (K). Moreover, the proper hereditarysaturated subsets of E are H = ∅ and H i = { v , . . . , v i } , for all i ≥
1. It follows thatthe admissible pairs ( H i , ∅ ) form a chain under the partial order of admissible pairs, and( E \ ( H i , ∅ )) satisfies the strong CSP for each i ≥
0. Hence, by Theorem 5.8, every ideal of L K ( E ) is completely irreducible.Next we examine the Leavitt path algebras where all completely irreducible ideals aregraded. Proposition 5.10.
The following are equivalent for any Leavitt path algebra L = L K ( E ) . Every completely irreducible ideal of L is graded. (2) Every prime ideal of L is graded. (3) Every ideal of L is graded. (4) The graph E satisfies condition (K).Proof. (1) ⇒ (3). By Proposition 5.1, every proper ideal of L is the intersection of completelyirreducible ideals. Thus if all completely irreducible ideals are graded, then so are all theother ideals of L (given that L itself is graded).(3) ⇒ (2). This is a tautology.(2) ⇒ (1). By Theorem 5.5, every non-graded completely irreducible ideal is a powerof a non-graded prime ideal. Thus, if all the prime ideals in L are graded, then so are thecompletely irreducible ideals.(3) ⇔ (4). See [1, Proposition 2.9.9].Let us now describe the Leavitt path algebras where every irreducible ideal is completelyirreducible. Proposition 5.11.
The following are equivalent for any Leavitt path algebra L = L K ( E ) . (1) Every irreducible ideal of L is completely irreducible. (2) Every graded prime ideal of L is completely irreducible. (3) Every ideal of L is graded, and every prime ideal of L is completely irreducible. (4) The graph E satisfies condition (K), and ( E \ ( H, B H )) satisfies the strong CSP foreach admissible pair ( H, B H ) such that E \ H is downward directed.Proof. (1) ⇒ (2). By Proposition 2.9, every graded prime ideal of L is irreducible, fromwhich the desired conclusion follows.(2) ⇒ (3). Suppose that (2) holds and that P is a non-graded prime ideal of L . Then,by Theorem 2.6, P = I ( H, B H ) + h f ( c ) i , where H = P ∩ E , c is a cycle without (K), E \ H = M ( s ( c )), and f ( x ) ∈ K [ x, x − ] is irreducible. By the same theorem, I ( H, B H )is a (graded) prime ideal, and so, by (2), I ( H, B H ) is completely irreducible. Hence, byTheorem 5.5, E \ ( H, B H ) satisfies condition (L), which contradicts c being a cycle without(K) in E . Therefore, if (2) holds, then all prime ideals of L must be graded, and so (3) mustalso hold, by Proposition 5.10.(3) ⇒ (4). Suppose that (3) holds. Then, by Proposition 5.10, E satisfies condition ( K ).Now let H be a hereditary saturated subset of E , such that E \ H is downward directed.Then P = I ( H, B H ) is a graded prime ideal of L , by Theorem 2.6. Since P is completelyirreducible, by hypothesis, Theorem 5.5 implies that ( E \ ( H, B H )) satisfies the strong CSP.(4) ⇒ (1). Suppose that (4) holds. Let us take an irreducible ideal I of L , and show thatit is completely irreducible. By Proposition 5.10, I must be graded, and so Proposition 2.9implies that I is a graded prime ideal. By Remark 5.6 and Theorem 2.6, we may assume that I = I ( H, B H ), where E \ H = ( E \ ( H, B H )) is downward directed. Since, by hypothesis, E satisfies condition (K), the quotient graph E \ ( H, B H ) must satisfy condition (L). It nowfollows from Theorem 5.5 that I is completely irreducible.18e conclude this section with an example of a graph E such that every irreducible idealof L K ( E ) is completely irreducible, but not every proper ideal is completely irreducible. Example 5.12.
Let E be the following row-finite graph. • v − o o • v − o o • v / / o o • v / / • v / / Clearly E satisfies condition (K), and it is easy to see that the hereditary saturated subsetsof E are precisely E , W + = { v i | i ≥ } , W − = { v i | i ≤ − } , and ∅ . Now, each of( E \ ( W + , ∅ )) = E \ W + and ( E \ ( W − , ∅ )) = E \ W − satisfies the strong CSP withrespect to itself, while E \ E and E \ ∅ are not downward directed. Thus, E satisfies theproperties in condition (4) of Proposition 5.11.On the other hand, the hereditary saturated subsets of E certainly do not form a chain,and so neither do the admissible pairs. Thus E does not satisfy condition (3) in Theo-rem 5.8. We conclude that in L K ( E ) every irreducible ideal is completely irreducible, butnot every proper ideal is completely irreducible. (Specifically, the zero ideal is not completelyirreducible, by Proposition 5.3.) In this section we characterize the ideals in an arbitrary Leavitt path algebra that can befactored as products (and intersections) of completely irreducible ideals. We also describe theLeavitt path algebras in which every ideal can be represented as a product (and intersection)of such ideals. We shall require the following observation.
Lemma 6.1. [3, Lemma 4.3] The following are equivalent for any Leavitt path algebra L = L K ( E ) and positive integer n . (1) The zero ideal is the (irredundant) intersection of n prime ideals. (2) The zero ideal is the (irredundant) intersection of n graded prime ideals. (3) E is the (irredundant) union of n maximal tails.Moreover, the maximal tails in (3) can be taken to be the complements in E of the setsof vertices contained in the prime ideals in (1) or (2). We are now ready for the last of our main results.
Theorem 6.2.
Let L = L K ( E ) be a Leavitt path algebra, and let I be a proper ideal of L .Then the following are equivalent. (1) I is the product of (finitely many) completely irreducible ideals. (2) I is the intersection of finitely many completely irreducible ideals. (3) I = I ( H, S ) + P ni =1 h f i ( c i ) i , where i ) 0 ≤ n is an integer, with n = 0 indicating that I = I ( H, S ) ; ( ii ) each c i is a cycle without exits in E \ ( H, S ) ; ( iii ) each f i ( x ) ∈ K [ x ] is a polynomial with a nonzero constant term; ( iv ) ( E \ ( H, S )) = S mi =1 M i is the irredundant union of finitely many maximal tails,with n ≤ m , such that, for each i ∈ { , . . . , n } we have s ( c i ) ∈ M i and s ( c i ) / ∈ M j for all j ∈ { , . . . , m } \ { i } , and for each i ∈ { n + 1 , . . . , m } every cycle withsource in M i has an exit and M i satisfies the strong CSP.Proof. (1) ⇒ (3) Suppose that we have I = P · · · P m for some completely irreducible ideals P , . . . , P m . By Theorem 2.1(1), I = I ( H, S ) + P i ∈ Y h f i ( c i ) i , where each c i is a cycle withoutexits in E \ ( H, S ), and each f i ( x ) ∈ K [ x ] is a polynomial with a nonzero constant term.Using Lemma 2.2(2) and the fact that a product of graded ideals is graded, we have I ( H, S ) = gr( I ) ⊆ m \ i =1 gr( P i ) = m Y i =1 gr( P i ) ⊆ gr( P · · · P m ) = gr( I ) , and hence I ( H, S ) = T mi =1 gr( P i ). By Theorem 5.5, Proposition 2.9, and Theorem 2.6, eachgr( P i ) is a prime ideal. Upon dropping terms, if needed, we may assume that the intersectionis irredundant. If I is non-graded, then, upon further reindexing, Lemma 4.7(2,4,5) givesthat | Y | = n ≤ m for some positive integer n , that for each i ∈ { , . . . , n } we have s ( c i ) / ∈ P i and s ( c i ) ∈ P j for all j ∈ { , . . . , m } \ { i } , and that( † ) I = n Y i =1 ( I ( H i , B H i ) + h f i ( c i ) i ) · m Y i = n +1 I ( H i , B H i ) , where gr( P i ) = I ( H i , B H i ) for each i . If I is graded, then I = I ( H, S ) = T mi =1 gr( P i ) andtaking n to be 0, we have I = Q mi = n +1 I ( H i , B H i ), by Lemma 2.2(2). Thus equation ( † )describes I in both situations. Now, by Lemma 4.8, I ( H i , B H i ) + h f i ( c i ) i is a product ofnon-graded prime ideals, for each i ∈ { , . . . , n } . Hence ( † ) gives a representation of I aproduct of prime-power ideals. Upon combining copies of the same prime ideal, droppingany redundant terms, and reindexing, we can apply Theorem 3.2 (and Theorem 5.5) toconclude that for each i ∈ { n + 1 , . . . , m } the ideal I ( H i , B H i ) must be one of the completelyirreducible ideals in the product I = P · · · P m .Since I ( H, S ) = T mi =1 gr( P i ), in the ring L/I ( H, S ) ∼ = L K ( E \ ( H, S )), by Lemma 6.1, wehave ( E \ ( H, S )) = S mi =1 M i , where each M i = ( E \ ( H, S )) \ (( E \ ( H, S )) ∩ ¯ Q i )is a maximal tail, the union is irredundant, and ¯ Q i = gr( P i ) /I ( H, S ). Then, for each i ∈ { , . . . , n } , we have s ( c i ) ∈ M i and s ( c i ) / ∈ M j for all j ∈ { , · · · , m } \ { i } . Moreover, foreach i ∈ { n + 1 , . . . , m } , the ideal ¯ Q i = I ( H i , B H i ) /I ( H, S ) is completely irreducible, andhence, by Theorem 5.5, every cycle with source in M i has an exit and M i satisfies the strongCSP. Thus the conditions in (3) hold for I .(3) ⇒ (1). Supposing that I satisfies (3), we have ( E \ ( H, S )) = S mi =1 M i , with the M i as in ( iv ). In L/I ( H, S ) ∼ = L K ( E \ ( H, S )), let H i = ( E \ ( H, S )) \ M i , and let P i be the ideal20f L containing I ( H, S ) such that ¯ P i = P i /I ( H, S ) = I ( H i , B H i ), for each i ∈ { , . . . , m } .Then, by hypothesis, c i / ∈ ¯ P i for i ∈ { , . . . , n } , and, by Theorem 2.6, each ¯ P i is a (graded)prime ideal. Moreover, by Theorem 5.5, ¯ P i is completely irreducible for i ∈ { n + 1 , . . . , m } .Also, by Lemma 6.1, { } = T mi =1 ¯ P i is an irredundant intersection in L/I ∼ = L K ( E \ ( H, S )).Then, using Lemma 2.2(2) and Lemma 4.7 (depending on whether or not I is graded), wehave( ‡ ) I/I ( H, S ) = n Y i =1 ( I ( H i , B H i ) + h f i ( c i ) i ) · m Y i = n +1 I ( H i , B H i ) . Now, for each i ∈ { , . . . , n } , let J i = P i + h f i ( c i ) i ⊆ L . Then, by Lemma 4.8, each ¯ J i isa product of non-graded prime ideals, and is hence is a product of completely irreducibleideals, by Theorem 5.5. Then equation ( ‡ ) implies that I = J · · · J n P n +1 · · · P m , which showsthat I is a product of completely irreducible ideals.The equivalence of (1) and (2) follows from Proposition 4.3 and Theorem 5.5.This theorem makes it easy to construct Leavitt path algebras having ideals that are orare not intersections (and products) of finitely many completely irreducible ideals. Example 6.3.
Let E be the following graph. • v + + (cid:7) (cid:7) ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ • v + + (cid:7) (cid:7) " " ❉❉❉❉❉❉❉❉ • v + + (cid:7) (cid:7) (cid:15) (cid:15) . . . • v Clearly, { v } is hereditary and saturated. Taking I = I ( { v } , ∅ ), we shall show that I is notthe intersection or product of any finite collection of completely irreducible ideals in L K ( E ).It is easy to see that E \ ( { v } , ∅ ) has the following form. • v + + (cid:7) (cid:7) • v + + (cid:7) (cid:7) • v + + (cid:7) (cid:7) . . . In this graph, the maximal tails are precisely the sets of the form { v i } ( i ≥ E \ ( { v } , ∅ )) cannot be expressed as the union of finitely many maximal tails. Therefore I is not the intersection or product of any finite collection of completely irreducible ideals in L K ( E ), by Theorem 6.2.Now let E to be the following graph. • v + + (cid:7) (cid:7) " " ❉❉❉❉❉❉❉❉ • v + + (cid:7) (cid:7) (cid:15) (cid:15) • v + + (cid:7) (cid:7) } } ③③③③③③③③ • v L K ( E ) the ideal I = I ( { v } , ∅ ) is a product andintersection of finitely many completely irreducible ideals.Our final goal is to classify the Leavitt path algebras in which every proper ideal is aproduct (and intersection) of finitely many completely irreducible ideals. Theorem 6.4.
The following are equivalent for any Leavitt path algebra L = L K ( E ) . (1) Every proper ideal of L is the product of (finitely many) completely irreducible ideals. (2) Every proper ideal of L is the intersection of finitely many completely irreducible ideals. (3) The graph E satisfies condition (K), and for every admissible pair ( H, S ) with H = E , ( E \ ( H, S )) is the union of finitely many maximal tails, each satisfying the strong CSP.Proof. (1) ⇔ (2). This follows from Theorem 6.2.(2) ⇒ (3). Suppose that (2) holds. Then, in particular, every irreducible ideal is theintersection of finitely many completely irreducible ideals, and so must actually be completelyirreducible. Thus, by Proposition 5.11, the graph E satisfies condition (K).Now, let ( H, S ) be an admissible pair, with H = E . Then, by hypothesis, I ( H, S )is the intersection of finitely many completely irreducible ideals. Thus, by Theorem 6.2,( E \ ( H, S )) is the union of finitely many maximal tails, each satisfying the strong CSP.(3) ⇒ (2). Suppose that (3) holds. Since the graph E satisfies condition (K), every idealof L is graded, by Proposition 5.10. Now, let I = I ( H, S ) be an arbitrary proper (graded)ideal of L . Since E satisfies condition ( K ), every cycle in E \ ( H, S ) has an exit. Thus, byTheorem 6.2, (3) implies that I is the intersection of finitely many completely irreducibleideals, giving the desired conclusion.For graphs with finitely many vertices, all the clauses in condition (3) of Theorem 6.4,except condition (K), are satisfied automatically, giving us the following result. Corollary 6.5.
Let E be a graph such that E is finite. Then every proper ideal of L K ( E ) is a product of completely irreducible ideals if and only if E satisfies condition (K). Let us further illustrate Theorem 6.4 by giving an example of a Leavitt path algebra whereevery proper ideal is a product of completely irreducible ideals, but may not be completelyirreducible itself.
Example 6.6.
Let E be the following graph (see Example 5.12). • v − o o • v − o o • v / / o o • v / / • v / / Then E satisfies condition (K), and the hereditary saturated subsets of E are precisely E , W + = { v i | i ≥ } , W − = { v i | i ≤ − } , and ∅ . Now,( E \ ( W + , ∅ )) = E \ W + = W − ∪ { v } and ( E \ ( W − , ∅ )) = E \ W − = W + ∪ { v } are maximal tails, each satisfying the strong CSP with respect to itself, and( E \ ( ∅ , ∅ )) = E = ( W − ∪ { v } ) ∪ ( W + ∪ { v } ) . It follows that E satisfies condition (3) in Theorem 6.4. On the other hand, as noted inExample 5.12, E does not satisfy condition (3) in Theorem 5.8. Thus in L K ( E ) every properideal is a product of completely irreducible ideals, but not every proper ideal is completelyirreducible. 22 cknowledgement We are grateful to the referee for a careful reading of the manuscript.
References [1] G. Abrams, P. Ara, and M. Siles Molina,
Leavitt path algebras,
Lecture Notes in Math-ematics , Springer-Verlag, London, 2017.[2] G. Abrams, J. Bell, and K. M. Rangaswamy,
On prime non-primitive von Neumannregular algebras,
Trans. Amer. Math. Soc. (2014) 2375–2392.[3] G. Abrams, Z. Mesyan, and K. M. Rangaswamy,
Products of ideals in Leavitt pathalgebras,
Comm. Algebra (2020) 1853–1871.[4] S. Aljohani, K. Radler, K. M. Rangaswamy, and A. Srivastava, Variations of primenessand factorizations of ideals in Leavitt path algebras, preprint.[5] S. Esin, M. Kanuni, A. Ko¸c, K. Radler, and K. M. Rangaswamy,
On Pr¨ufer-like prop-erties of Leavitt path algebras,
J. Algebra Appl. (2020) 2050122.[6] S. Esin, M. Kanuni, and K. M. Rangaswamy, On intersections of two-sided ideals ofLeavitt path algebras,
J. Pure Appl. Algebra (2017) 632–644.[7] L. Fuchs, W. J. Heinzer, and B. Olberding,
Commutative ideal theory without finitenessconditions: Primal ideals,
Trans. Amer. Math. Soc. (2004) 2771–2798.[8] L. Fuchs, W. J. Heinzer, and B. Olberding,
Commutative ideal theory without finitenessconditions: Completely irreducible ideals,
Trans. Amer. Math. Soc. (2006) 3113–3131.[9] L. Fuchs and F. Mosteig,
Ideal theory in Pr¨ufer domains–an unconventional approach,
J. Algebra (2002) 411–430.[10] W. Heinzer and B. Olberding,
Unique irredundant intersections of completely irreducibleideals,
J. Algebra (2005) 432–448.[11] T. van den Hove,
The arithmetic of ideals in Leavitt path algebras, bachelor’s thesis,Ghent University, 2019.[12] K. M. Rangaswamy,
Theory of prime ideals of Leavitt path algebras over arbitrary graphs,
J. Algebra (2013) 73–96.[13] K. M. Rangaswamy,
On generators of two-sided ideals of Leavitt path algebras overarbitrary graphs,
Comm. Algebra (2014) 2859–2868.[14] K. M. Rangaswamy, The multiplicative ideal theory of Leavitt path algebras,
J. Algebra487