Relative polynomial closure and monadically Krull monoids of integer-valued polynomials
aa r X i v : . [ m a t h . N T ] M a r RELATIVE POLYNOMIAL CLOSURE AND MONADICALLYKRULL MONOIDS OF INTEGER-VALUED POLYNOMIALS
SOPHIE FRISCH
Abstract.
Let D be a Krull domain and Int( D ) the ring of integer-valuedpolynomials on D . For any f ∈ Int( D ), we explicitly construct a divisor homo-morphism from [[ f ]], the divisor-closed submonoid of Int( D ) generated by f , toa finite sum of copies of ( N , +). This implies that [[ f ]] is a Krull monoid.For V a discrete valuation domain, we give explicit divisor theories of varioussubmonoids of Int( V ). In the process, we modify the concept of polynomial clo-sure in such a way that every subset of D has a finite polynomially dense subset.The results generalize to Int( S, V ), the ring of integer-valued polynomials on asubset, provided S doesn’t have isolated points in v -adic topology. Introduction
The ring of integer-valued polynomials Int( Z ) enjoys quite chaotic non-uniquefactorization, in the following sense: given any finite list of natural numbers 1 S, D ). If S doesn’t have any isolated points in any of the topologiesgiven by essential valuations of D , we can construct a divisor homomorphism from[[ f ]] to a finite direct sum of copies of ( N , +) [Theorem 5.4]. This implies that [[ f ]]is a Krull monoid, and hence, that Int( S, D ) is monadically Krull.In the special case where D is a discrete valuation domain, we can actuallydetermine the divisor theories of certain submonoids of Int( S, D ) [Proposition 4.2and Theorem 5.3]. Mathematics Subject Classification. Primary 13F20; Secondary 20M13, 13A05, 13B25,11C08, 11R09. Key words and phrases. monoid, factorization, monadically Krull, divisor homomorphism,divisor theory, integer-valued polynomial, polynomial closure.This publication was supported by the Austrian Science Fund FWF, grant P23245-N18. For the purpose of constructing divisor homomorphisms on monoids of integer-valued polynomials, we will study “relative” polynomial closure, that is, polyno-mial closure with respect to a subset of K [ x ], in section 2. This modification of theconcept of polynomial closure makes it possible to find finite polynomially densesubsets of arbitrary sets in section 3. Equipped with these finite polynomiallydense sets we construct the actual divisor homomorphisms and, in some cases,divisor theories, to finite sums of copies of ( N , +) in sections 4 and 5.A short review of integer-valued polynomial terminology: Let D be a domainwith quotient field K , S ⊆ K and f ∈ K [ x ]. f is called integer-valued if f ( D ) ⊆ D and f is called integer-valued on S if f ( S ) ⊆ D . If there are several possibilitiesfor D , we say D -valued on S instead of integer-valued on S .The ring of integer-valued polynomials on D is written Int( D ), and the ring ofinteger-valued polynomials on a subset S of the quotient field of D is denoted byInt( S, D ): Int( S, D ) = { f ∈ K [ x ] | f ( S ) ⊆ D } , Int( D ) = Int( D, D ) . Definition 1.1. Let D be a domain with quotient field K , S ⊆ D and f ∈ Int( S, D ). The divisor-closed submonoid of Int( S, D ) generated by f , which wewrite [[ f ]], is the multiplicative monoid consisting of all g ∈ Int( S, D ) for whichthere exists m ∈ N and h ∈ Int( S, D ), such that g · h = f m .Keep in mind that the elements of [[ f ]] are not just polynomials in Int( S, D ) thatdivide some power of f in K [ x ]. The co-factor is also required to be in Int( S, D ).We will frequently use the following divisibility criterion for [[ f ]]. Remark . Let [[ f ]] be the divisor closed submonoid of Int( S, D ) as in Definition1 . a, b ∈ [[ f ]]. Then a divides b in [[ f ]] if and only if a divides b in K [ x ] andthe cofactor c = b/a is in Int( S, D ).Multiplying a polynomial in [[ f ]] by a constant in D does not in general resultin an element of [[ f ]]. We can multiply elements of [[ f ]] by some suitable constants,though. Lemma 1.3. Let V be the valuation domain of a valuation v on K , S ⊆ V , f ∈ Int( S, V ) and [[ f ]] the divisor-closed submonoid of Int( S, V ) generated by f .Let g ∈ [[ f ]] and a ∈ K . If − min s ∈ S v ( g ( s )) ≤ v ( a ) ≤ then ag ∈ [[ f ]] .Proof. Let g, h ∈ Int( S, V ) and m ∈ N such that gh = f m . Then both ag and a − h are in Int( S, V ), and ag · a − h = f m . (cid:3) We recall the definitions of ideal content and fixed divisor, whose interplay willbe an important ingredient of proofs. Let R be a domain and f ∈ R [ x ]. Thecontent of f , denoted c ( f ), is the fractional ideal generated by the coefficients of f . If R is a principal ideal domain, we identify, by abuse of notation, ideals bytheir generators and say that c ( f ) is the gcd of the coefficients of f . A polynomial f ∈ R [ x ] is called primitive if c ( f ) = R , that is, in the case of a PID, if c ( f ) = 1. ONOIDS OF INTEGER-VALUED POLYNOMIALS 3 Definition 1.4. Let D be a domain with quotient field K , S ⊆ D and f ∈ K [ x ] \ { } . The fixed divisor of f on S , denoted d S ( f ), is the D -submodule of K generated by the image f ( S ). Note that d S ( f ) is a fractional ideal. If S = D , wewrite d( f ) for d D ( f ). If D is a PID, we will, by abuse of notation, sometimes writea generator to stand for the ideal, e.g., d S ( f ) = 1 for d S ( f ) = D . A polynomial f ∈ Int( S, D ) is called image-primitive if d S ( f ) = D .For polynomials in D [ x ], image-primitive implies primitive, but not vice versa.One difference between ideal content and fixed divisor is that the ideal content ismultiplicative for sufficiently nice rings (called Gaussian rings), including principalideal rings, whereas the fixed divisor is not multiplicative. d S ( f ) d S ( g ) containsd S ( f g ), but the containment can be strict. Remark . Two easy but useful facts:(1) If f ∈ Int( S, D ) is image-primitive then f n is image-primitive for all n ∈ N .(2) If f ∈ Int( S, D ) is image-primitive then all divisors in Int( S, D ) of f arealso image-primitive. Remark . In case D is an intersection of valuation rings, then every f ∈ Int( S, D ) is also in Int( S, V ) for all these valuation rings, and f may be image-primitive as an element of Int( S, V ), but not as an element of Int( S, D ). In thiscase, we write v ( f ( S )) := min s ∈ S v ( f ( s ))and write v ( f ( S )) = 0 to express that f is image-primitive when regarded as anelement of Int( S, V ).Regarding valuation terminology: we use additive valuations, that is, a valuationis a map v : K \ { } → Γ, where (Γ , +) is a totally ordered group, satisfying(1) v ( ab ) = v ( a ) + v ( b )(2) v ( a + b ) ≥ min( v ( a ) , v ( b ))and we set v (0) = ∞ . The valuation ring of a valuation v on a field K is V = { k ∈ K | v ( k ) ≥ } and the valuation group is the image of v in Γ.2. Relative polynomial closure Definition 2.1 (relative polynomial closure) . Fix a domain D with quotient field K . Let T ⊆ K and F ⊆ K [ x ].The polynomial closure of T relative to F is C F ( T ) = { s ∈ K | ∀ f ∈ F ∩ Int( T, D ) : f ( s ) ∈ D } . If T ⊆ S ⊆ K , and C F ( T ) ⊇ S we call T polynomially dense in S relative to F .The definition of polynomial closure and polynomial density depends on thechoice of D . If there is any doubt about D , we say D -polynomial closure and D -polynomially dense. ONOIDS OF INTEGER-VALUED POLYNOMIALS 4 Polynomial closure relative to K [ x ] is the “usual” polynomial closure, introducedby Gilmer [6] and studied by McQuillan [7], the present author [3], Cahen [1], Parkand Tartarone [8] and Chabert [2], among others. The reason why we generalizethe well-known concept of polynomial closure will become apparent in the nextsection: when we consider polynomial closure relative to a set of polynomials whoseirreducible factors are restricted to a finite set, it becomes possible to find finitepolynomially dense subsets of any fractional set. Remark . The following properties of polynomial closure relative to a subset F of K [ x ] are easy to check.(1) C F ( T ) = T f ∈F∩ Int( T,D ) f − ( D )(2) Polynomial closure relative to F is a closure operator, in the sense that(a) T ⊆ C F ( T )(b) C F ( C F ( T )) = C F ( T )(c) T ⊆ S = ⇒ C F ( T ) ⊆ C F ( S )(3) Polynomial closure relative to F is the closure given by a Galois correspon-dence that maps every subset T of K to a subset of F , and every subset G of F to a subset of K , namely, T 7→ F ∩ Int( T, D ) and G \ f ∈ G f − ( D ) . (4) If F ⊆ F ⊆ K [ x ] then C F ( T ) ⊆ C F ( T ).(5) If T is polynomially dense in S relative to F , and F ⊆ F , then T ispolynomially dense in S relative to F .When the domain D is a valuation ring, then polynomially dense subsets of S relative to F are easily characterized: Lemma 2.3. Let v be a valuation on a field K , V its valuation ring, T ⊆ S ⊆ K and F ⊆ K [ x ] . Consider (1) ∀ f ∈ F min t ∈ T v ( f ( t )) = min s ∈ S v ( f ( s ))(2) T is V -polynomially dense in S relative to F . (1) implies (2) . If F is closed under multiplication by non-zero constants in K then (2) implies (1) .Proof. (1 ⇒ 2) For every polynomial f ∈ F ∩ Int( T, V ), min t ∈ T v ( f ( t )) ≥ s ∈ S v ( f ( s )) ≥ f ∈ Int( S, V ).(2 ⇒ 1) For every f ∈ F , min t ∈ T v ( f ( t )) ≥ min s ∈ S v ( f ( s )), since T ⊆ S . If f ∈ F and α ∈ Z are such that min t ∈ T v ( f ( t )) ≥ α > min s ∈ S v ( f ( s )), pick a ∈ K with v ( a ) = − α . Then af ∈ F ∩ Int( T, V ), but af Int( S, V ), so T is not V -polynomially dense in S relative to F . (cid:3) ONOIDS OF INTEGER-VALUED POLYNOMIALS 5 Finite polynomially dense subsets Let F be a finite set of irreducible polynomials in K [ x ] and F the multiplicativesubmonoid of K [ x ] generated by F and the non-zero constants of K . That is, F consists of all non-zero polynomials in K [ x ] whose irreducible factors in K [ x ] are(up to multiplication by non-zero constants) in F .We will now construct, for every subset S of a discrete valuation ring V , a finitepolynomially dense subset of S relative to F . It is possible to admit fractionalsubsets of K , but for simplicity’s sake we restrict ourselves to subsets of V .By discrete valuation, we mean, more precisely, a discrete rank 1 valuation, thatis, a valuation v whose value group is isomorphic to Z . A normalized discretevaluation is one whose value group is actually equal to Z . The valuation ring ofa discrete valuation is called discrete valuation ring, abbreviated DVR. As we allknow, a DVR is a local principal ideal domain. Remark . Let v be a discrete valuation on K with valuation ring V , f ∈ K [ x ],and L ⊇ K a finite-dimensional field extension over which f splits. Let w be anextension of v to L ( w | K = v ), W the valuation ring of w and P its maximal ideal.Say f splits as f ( x ) = c Q kj =1 ( x − b j ) Q mj =1 ( x − a j ) with w ( b j ) < ≤ j ≤ k and w ( a j ) ≥ ≤ j ≤ m over L .Then for all s ∈ V , v ( f ( s )) = w ( c ) + k X j =1 w ( b j ) + m X j =1 w ( s − a j ) Proof. This follows from the fact that w ( s ± b ) = w ( b ) whenever w ( b ) < w ( s ). (cid:3) Definition 3.2. Let X be a topological space and S ⊆ X . An isolated point of S is an element s ∈ X having a neighborhood U such that U ∩ S = { s } . Proposition 3.3. Let v be a discrete valuation on K and V its valuation ring.Let F = ∅ be a finite set of monic irreducible polynomials in K [ x ] and F the setof those polynomials in K [ x ] whose monic irreducible factors are all in F . Let S ⊆ V . (1) Then there exists a finite subset T ⊆ S such that ∀ f ∈ F min t ∈ T v ( f ( t )) = min s ∈ S ( v ( f ( s ))) and every such T ⊆ S is, in particular, a finite set that is polynomiallydense in S relative to F . (2) If no root of any f ∈ F is an isolated point of S in v -adic topology, then theabove set T can be chosen such as not to contain any root of any f ∈ F . (3) Let L be the splitting field of F over K , w an extension of v to L and W the valuation ring of w . Let A be the set of distinct roots of polynomials of F in W . Then T in (1) and (2) can be chosen with | T | ≤ max(1 , | A | ) . ONOIDS OF INTEGER-VALUED POLYNOMIALS 6 Proof. Let L , w , W , and A as in (3). Let P be the maximal ideal of W . Wecall the elements of A “the roots”. We may assume S = ∅ and A = ∅ (otherwisethe claimed facts are trivial). In view of Remark 3.1 it suffices to construct a set T ⊆ S such that, for every finite sequence ( a i ) mi =1 in A ,min t ∈ T m X i =1 w ( t − a i ) = min s ∈ S m X i =1 w ( s − a i )We will do this by constructing a finite covering C of S by disjoint sets C ⊆ W andfor each C ∈ C choosing a representative t ∈ C ∩ S such that w ( t − a ) ≤ w ( s − a )for every a ∈ A and every s ∈ C ∩ S . This representative t ∈ C ∩ S then satisfies ∀ f ∈ F v ( f ( t )) = min s ∈ C ∩ S v ( f ( s )), by Remark 3.1. If we take T to be the setof representatives of covering sets C ∈ C then for every f ∈ F , min s ∈ S v ( f ( s )) isrealized by some s ∈ T . By Lemma 2.3, this makes T polynomially dense in S relative to F .For any ideal I of W , we call a residue class r + I “relevant” if S ∩ ( r + I ) = ∅ .We construct C , C n ( n ≥ 0) and T inductively. Before step 0, initialize T = ∅ , C = ∅ , C = { W } .At the beginning of step n , C is a finite set of relevant residue classes of various P k with k < n while C n is a finite set of relevant residue classes of P n eachcontaining at least one root. In step n , initialize C n +1 = ∅ ; then go through each C ∈ C n and process it as follows:(1) If C ∩ S = { c } with c ∈ A then put c in T and C in C . Note that in thiscase C ∩ V is a v -adic neighborhood of c whose intersection with S is { c } ,and that therefore c ∈ A is an isolated point of S .(2) Else, if C contains a relevant residue class D of P n +1 which doesn’t containa root, pick such a D , add a representative of D ∩ S to T ; then put C in C .(3) Else place all relevant residue classes of P n +1 contained in C (each con-taining a root, by construction) in C n +1 .If C n +1 is empty at the end of step n , stop. Otherwise proceed to step n + 1.Note that after each step n , C ∪ C n +1 is a covering of S . When the algorithmterminates with C n +1 = ∅ , then C is a covering of S and T contains for each C ∈ C a representative t ∈ C ∩ S satisfying w ( t − a ) = min s ∈ C ∩ S w ( s − a ) for all a ∈ A .Therefore v ( f ( t )) = min s ∈ C ∩ S v ( f ( s )) for all f ∈ F by Remark 3.1.The algorithm terminates when no root is left in S C n +1 . For each root a ∈ A ,one can give an upper bound on n such that a is no longer in C n +1 . Namely, let n such that w ( a − a ′ ) < n for all roots a = a ′ . If ( a + P n +1 ) ∩ S = ∅ then aresidue class containing a has been dropped as not relevant at or before step n ,so a + P n +1 6∈ C n +1 . If ( a + P n +1 ) ∩ S = { a } , then a residue class containing a isplaced in C at step n + 1 or earlier. Otherwise, a + P n +1 contains an element of S other than a . Let s ∈ ( a + P n +1 ) ∩ S , with w ( s − a ) = m minimal. Then a + P m will be placed in C by step m . ONOIDS OF INTEGER-VALUED POLYNOMIALS 7 This shows (1). For (2), note that the set T thus constructed contains no rootof any f ∈ F except such as are isolated points of S in v -adic topology. For (3),note that every time an element is added to T , a set containing at least one rootis transferred from C n to C and the number of roots in S C ∈C n C decreases. (cid:3) Remark . Thanks to the anonymous referee for pointing out that parts (1) and(2) of Proposition 3.3 can be show more quickly by applying Dickson’s theorem[5][Thm. 1.5.3], which says that the set of minimal elements of any subset N of N m is finite and that for every a ∈ N there exists a minimal element b ∈ N with b ≤ a , to the subset N = { ( w ( s − a )) a ∈ A | s ∈ S } of N A .4. Divisor theories for monoids of integer-valued polynomials ondiscrete valuation rings A short review of monoid terminology used in the definition of divisor homomor-phism: By monoid we mean a semigroup that has a neutral element. All monoidswe consider here are commutative, and they are cancellative, that is, whenever ab = cb or ba = bc , it follows that a = c .Let ( M, +) be a commutative monoid, written additively, and a, b ∈ M .(1) We say that a divides b in M , and write a | b , whenever there exists c ∈ M such that a + c = b .(2) We call an element d ∈ M a greatest common divisor , abbreviated gcd, ofa subset A ⊆ M , if(a) d | a for all a ∈ A (b) for all c ∈ M : if c | a for all a ∈ A then c | d . Definition 4.1. A monoid homomorphism ϕ : G → H is called a divisor homo-morphism if ϕ ( a ) | ϕ ( b ) in H implies a | b in G .A divisor homomorphism ϕ : G → P ni =1 ( N , +) is called a divisor theory if eachof the basis vectors e i (having 1 in the i -th coordinate and zeros elsewhere) occursas gcd of a finite set of images ϕ ( g ).We are preparing to construct divisor homomorphisms from certain submonoidsof Int( S, D ), where D is a Krull domain, to finite sums of copies of ( N , +), relatingdivisibility in Int( S, D ) to divisibility in a finitely generated free commutativemonoid, which a priori looks much simpler. If ( M, +) is a direct sum of k copiesof ( N , +), then the divisibility relation in M is just the partial order given bythe order relations on each component: Let a, b ∈ M with a = ( a , . . . , a k ) and b = ( b , . . . , b k ). Then a | b in M is equivalent to a i ≤ b i for all 1 ≤ i ≤ k .Therefore, any set { ( m i , m i , . . . , m ik ) | i ∈ I } of elements of M has a unique gcd,namely, d = (min i ( m i ) , min i ( m i ) , . . . , min i ( m ik )).In what follows, we denote the normalized discrete valuation on K ( x ) corre-sponding to an irreducible polynomial h ∈ K [ x ] by v h ; for g ∈ K [ x ], v h ( g ) is theexponent to which h occurs in the essentially unique factorization of g in K [ x ] intoirreducible polynomials, and for g /g ∈ K ( x ), v h ( g /g ) = v h ( g ) − v h ( g ). ONOIDS OF INTEGER-VALUED POLYNOMIALS 8 In this section we examine the special case Int( S, V ), where V is a discretevaluation ring (DVR). Proposition 4.2. Let v be a normalized discrete valuation on K and V its valua-tion ring. Let H be a finite set of pairwise non-associated irreducible polynomialsin K [ x ] and H the multiplicative submonoid of K [ x ] generated by H and the non-zero constants in K . Let S ⊆ V such that no root of any h ∈ H is an isolatedpoint of S in v -adic topology. Let F = H ∩ Int( S, V ) .There exists a finite subset T of S that is polynomially dense in S relative to H and contains no root of any h ∈ H ; and for every such Tϕ : F → X h ∈ H ( N , +) ⊕ X t ∈ T ( N , +) , ϕ ( g ) = (( v h ( g ) | h ∈ H ) , ( v ( g ( t )) | t ∈ T )) , is a divisor homomorphism. If T is chosen minimal, ϕ is a divisor theory.Proof. The existence of a finite polynomially dense subset T containing no root ofany h ∈ H is Proposition 3.3. Once we have a finite dense set, a minimal denseset can be obtained by removing redundant elements. ϕ is clearly a monoid homomorphism. Now suppose a, b ∈ F such that ϕ ( a ) | ϕ ( b ), and set c = b/a . We must show c ∈ Int( S, V ). ϕ ( a ) | ϕ ( b ) means v h ( a ) ≤ v h ( b ) for all h ∈ H and v ( a ( t )) ≤ v ( b ( t )) for all t ∈ T .The first shows c ∈ K [ x ], and therefore c ∈ H , and the second shows that c ( t ) ∈ V for all t ∈ T . Since T is polynomially dense in S relative to H , it follows that c ∈ Int( S, V ). We have shown ϕ to be a divisor homomorphism.It remains to show that every e h for any h ∈ H and every e t for any t ∈ T occursas the gcd of a finite set of images of elements of F , provided T is minimal.We may assume, without changing H , F or ϕ in any way, that the elements of H are in V [ x ] and primitive.First, let p be a generator of the maximal ideal of V . The constant polynomial p is an element of F satisfying v h ( p ) = 0 for all h ∈ H and v ( p ( t )) = 1 for all t ∈ T .Second, we note that every polynomial h ∈ H is an element of F satisfying v h ( h ) = 1 and v l ( h ) = 0 for every l ∈ H \ { h } .Third, we show that for every t ∈ T , there exists g t ∈ F such that v ( g t ( t )) = 0and v ( g t ( r )) > r ∈ T \ { t } . We use the minimality of T and Lemma 2.3:Since T is polynomially dense in S relative to H , but T \ { t } is not, there exists apolynomial k ∈ H with v ( k ( t )) = min s ∈ S v ( k ( s )) and v ( k ( r )) > min s ∈ S v ( k ( s )) forall r ∈ T \ { t } . Let k be such a polynomial and α = v ( k ( t )). Then g t ( x ) = p − α k ( x )has the desired properties.Fourth, we show that for every t ∈ T and h ∈ H there exists g th ∈ F such that v ( g th ( t )) = 0 and v h ( g th ) > 0. Let k be any polynomial in F with v h ( k ) > 0. If v ( k ( t )) = α > 0, set g th ( x ) = p − α k ( x ) g t ( x ) α . ONOIDS OF INTEGER-VALUED POLYNOMIALS 9 Now for any h ∈ H and t ∈ T , e h = gcd( { ϕ ( g th ) | t ∈ T } ∪ { ϕ ( h ) } ) and e t = gcd( { ϕ ( g r ) | r = t } ∪ { ϕ ( p ) } ) . (cid:3) Divisor homomorphisms on monadic monoids of integer-valuedpolynomials What we have found out about the submonoid of Int( S, D ) consisting of polyno-mials whose irreducible factors in K [ x ] come from a fixed finite set, we now applyto the divisor closed submonoid of Int( S, D ) generated by a single polynomial.Recall that [[ f ]], the divisor-closed submonoid of Int( S, D ) generated by f , is themultiplicative monoid consisting of all those g ∈ Int( S, D ) which divide some powerof f in Int( S, D ). Also, it will be useful to recall the definition of image-primitive,and of d S ( f ), the fixed divisor of f on S from Definition 1.4.First let us get a trivial case out of the way: Lemma 5.1. Let V be a DVR, S ⊆ V and f ∈ V [ x ] with d S ( f ) = V . Let F ⊆ V [ x ] be a set of primitive polynomials in V [ x ] representing the differentirreducible factors of f in K [ x ] . Let F be the multiplicative submonoid of V [ x ] generated by F and the units of V . Then (1) [[ f ]] = F (2) Every element g of [[ f ]] is in V [ x ] , is primitive, and satisfies d S ( g ) = V . (3) If g, h ∈ [[ f ]] , then g divides h in [[ f ]] if and only if g divides h in K [ x ] . (4) ϕ : [[ f ]] → P h ∈ F ( N , +) , ϕ ( g ) = ( v h ( g ) | h ∈ F ) , is a divisor theory.Proof. We will show (1) and (2). The remaining statements follow from (1). f ∈ V [ x ] is image-primitive on S and hence primitive. The same holds for allpowers of f and for all divisors in V [ x ] of any power of f by Remark 1.5. Thereforeevery divisor in V [ x ] of any power of f is in F , and vice versa, every element of F is a divisor in V [ x ] of some power of f . Therefore every element of F isimage-primitive on S , and also F ⊆ [[ f ]].Now let g ∈ [[ f ]]. Let m ∈ N and h ∈ Int( S, V ) with hg = f m . Then h = c ˜ h and g = d ˜ g with ˜ g, ˜ h ∈ F and c, d ∈ K . Since ˜ g and ˜ h are image-primitive on S , wemust have v ( c ) ≥ v ( d ) ≥ 0. Since f m is primitive, v ( c ) = − v ( d ). It followsthat v ( c ) = v ( d ) = 0 and therefore g, h ∈ F . (cid:3) Let D be a domain with quotient field K , S a subset of D , and f ∈ Int( S, D ).Let H be a set of representatives (up to multiplication by a non-zero constant)of the irreducible factors of f in K [ x ]. For instance, H could be the set of monicirreducible factors of f in K [ x ]. Or, in case that D is a principal ideal domain,such as, for instance, a discrete valuation domain, H can be chosen to be the setof primitive irreducible polynomials in D [ x ] dividing f in K [ x ]. By H we denotethe multiplicative submonoid of K [ x ] \ { } generated by H and the constants in K \ { } . (Note that H depends only on f , not on the choice of H ). Obviously ONOIDS OF INTEGER-VALUED POLYNOMIALS 10 [[ f ]] ⊆ H ∩ Int( S, D ). We now examine when the equality holds. In this non-trivialcase we can give a divisor theory of [[ f ]] [Theorem 5.3]. Otherwise, we have to becontent with a divisor homomorphism [Theorem 5.4]. Theorem 5.2. Let V be a discrete valuation domain with quotient field K , S ⊆ V and f ∈ Int( S, V ) . Let H be multiplicative submonoid of K [ x ] generated by theirreducible factors of f and the non-zero constants. If d S ( f ) = V then [[ f ]] = H ∩ Int( S, V ) .Proof. Let H be the set of primitive irreducible polynomials in V [ x ] that divide f in K [ x ]. Let f = c ( f ) ˜ f with c ( f ) ∈ K \ { } the content of f and ˜ f ∈ V [ x ]primitive. For arbitrary b ∈ V \ { } , we show that b ˜ f ∈ [[ f ]].Let b ∈ V \ { } . Since d S ( f ) = V , v (d S ( f )) > m ∈ N such that mv (d S ( f )) ≥ v ( b ) − v ( c ( f )).Then f m +1 = ( f m c ( f ) b − ) b ˜ f , and both ( f m c ( f ) b − ) and b ˜ f are in Int( S, V ).Therefore b ˜ f ∈ [[ f ]].Now that b ˜ f ∈ Int( S, V ) for arbitrary b ∈ V \ { } , all factors of b ˜ f in V [ x ] are in[[ f ]]. Therefore, all primitive irreducible factors of f and all non-zero constants of V , and furthermore, all products of such elements, are in [[ f ]]. Finally, by Lemma1.3, we can multiply elements of [[ f ]] by any constant a ∈ K with v ( a ) < 0, as longas the result is integer-valued on S . Therefore, H ∩ Int( S, V ) ⊆ [[ f ]].The reverse inclusion [[ f ]] ⊆ H ∩ Int( S, V ) is trivial. (cid:3) Theorem 5.3. Let v be a normalized discrete valuation on K and V its valuationring. Let S ⊆ V and f ∈ Int( S, V ) , such that no root of f is an isolated pointof S in v -adic topology. Let H be the set of different monic irreducible factors of f in K [ x ] and H the multiplicative submonoid of K [ x ] generated by H and thenon-zero constants in K . By [[ f ]] denote the divisor-closed submonoid of Int( S, V ) generated by f .There exists a finite polynomially dense subset T of S relative to H that doesnot contain any root of f ; and for every such Tϕ : [[ f ]] → X h ∈ H ( N , +) ⊕ X t ∈ T ( N , +) ϕ ( g ) = (( v h ( g ) | h ∈ H ) , ( v ( g ( t )) | t ∈ T )) , is a divisor homomorphism. If d S ( f ) = V and T is chosen minimal then ϕ is adivisor theory.Proof. [[ f ]] is a submonoid of F = H ∩ Int( S, V ). The monoid homomorphism ϕ in the theorem is the restriction of the divisor homomorphism of Proposition 4.2to F and therefore itself a divisor homomorphism. If d S ( f ) = V then [[ f ]] = F byTheorem 5.2. In this case, ϕ is a divisor theory by Proposition 4.2, provided T isminimal. (cid:3) ONOIDS OF INTEGER-VALUED POLYNOMIALS 11 Recall that a Krull domain R is a domain satisfying the following conditionswith respect to Spec ( R ), the set of prime ideals of height 1:(1) For every P ∈ Spec ( R ), the localization R P is a DVR.(2) R = T P ∈ Spec ( R ) R P (3) Each non-zero r ∈ R lies in only finitely many P ∈ Spec ( R ).If R is a Krull domain, we denote the normalized discrete valuation on thequotient field of R whose valuation ring is R P by v P . Such a valuation is called anessential valuation of the Krull domain R .Again, the existence of finite D P -polynomially dense subsets of S relative to F in the following theorem is guaranteed by Proposition 3.3. Theorem 5.4. Let D be a Krull domain with quotient field K and S ⊆ D suchthat S doesn’t have any isolated points in v -adic topology for any essential valuation v of D . Let f ∈ Int( S, D ) , and [[ f ]] the divisor-closed multiplicative submonoid of Int( S, D ) generated by f .Let H be the finite set of different monic irreducible factors of f in K [ x ] and H the multiplicative submonoid of K [ x ] generated by H and the non-zero constants.Let P be the finite set of primes P of height of D such that either f D P [ x ] or f ∈ D P [ x ] and v P ( f ( S )) > . For each P ∈ P , let T P be a finite subset of S that is D P -polynomially dense relative to H in S and contains no root of f .Let ( M, +) = X h ∈ H ( N , +) ⊕ X P ∈P X t ∈ T P ( N , +) . Then ϕ : [[ f ]] → M, ϕ ( g ) = (( v h ( g ) | h ∈ H ) , ((v P ( g ( t )) | t ∈ T P ) | P ∈ P )) , is a divisor homomorphism.Proof. It is clear that ϕ is a monoid homomorphism. Now assume a, b ∈ [[ f ]] with ϕ ( a ) | ϕ ( b ). It suffices to show that a divides b in K [ x ] and that the co-factor c = b/a is in Int( S, D P ) for all P ∈ Spec ( D ), because then c ∈ Int( S, D ), whichimplies that a divides b in [[ f ]] by Remark 1.2.Let c = b/a . That c is in K [ x ] follows from v h ( a ) ≤ v h ( b ) for all irreduciblefactors h of a and b in K [ x ].Consider a prime P of height 1 of D that is not in P . For such a prime, f ∈ D P [ x ]and f is image-primitive in Int( S, D P ). We may apply Lemma 5.1 (3) and deducethat c ∈ Int( S, D P ).Now for P ∈ P , let ψ P be the projection of M onto P h ∈ H ( N , +) ⊕ P t ∈ T P ( N , +),and call the latter monoid M ( P ). From ϕ ( a ) | ϕ ( b ) it follows that ψ P ( ϕ ( a )) divides ψ P ( ϕ ( b )). Let [[ f ]] P be the divisor closed submonoid of Int( S, D P ) generated by f . Then [[ f ]] is a submonoid of [[ f ]] P , and ψ P ◦ ϕ is the restriction to [[ f ]] of thedivisor homomorphism in 4.2. Now the fact that ψ P ( ϕ ( a )) divides ψ P ( ϕ ( b )) implies c ∈ Int( S, D P ), by Proposition 4 . (cid:3) ONOIDS OF INTEGER-VALUED POLYNOMIALS 12 Corollary 5.5. Let D be a Krull domain and S a subset that doesn’t have anyisolated points in any of the topologies given by essential valuations of D . Let f ∈ Int( S, D ) . Then [[ f ]] , the divisor closed submonoid of Int( S, D ) generated by f , is a Krull monoid.In particular, for every Krull domain D and every f ∈ Int( D ) , the divisor closedsubmonoid [[ f ]] of Int( D ) generated by f is a Krull monoid.Proof. Indeed, the existence of a divisor homomorphism from [[ f ]] to a finite sumof copies of ( N , +) in Theorem 5.4 ensures that [[ f ]] is a Krull monoid, see[5][Thm. 2.4.8]. (cid:3) Monoids with the property that the divisor closed submonoid generated by anysingle element is a Krull monoid have been called monadically Krull by A. Rein-hart. Without using divisor homomorphisms, through an approach completelydifferent from ours, Reinhart showed that Int( D ) is monadically Krull whenever D is a principal ideal domain [9][Thm. 5.2].Corollary 5.5 generalizes Reinhart’s result to Krull domains, and to integer-valued polynomials on sufficiently nice subsets. The explicit divisor homomor-phisms of Proposition 4.2 and Theorems 5.3 and 5.4 give additional informationon the arithmetic of submonoids of Int( D ). It remains an open problem to find theprecise divisor theories (cf. Def. 4.1) of those monoids of integer-valued polynomialsfor which the above theorems provide divisor homomorphisms. Acknowledgment. Enthusiastic thanks go out to the anonymous referee forhis/her meticulous reading of the paper and the resulting corrections. References [1] P.-J. 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Reinhart , On monoids and domains whose monadic submonoids are Krull , in Commu-tative Algebra – Recent Advances in Commutative Rings, Integer-valued Polynomials, andPolynomial Functions, M. Fontana, S. Frisch, and S. Glaz, eds., Springer, 2014. Institut f¨ur Mathematik A, Technische Universit¨at Graz, Steyrergasse 30,8010 Graz, Austria E-mail address ::