\mathfrak{X}-elements in multiplicative lattices -- A generalization of J-ideals, n-ideals and r-ideals in rings
aa r X i v : . [ m a t h . A C ] J a n X -ELEMENTS IN MULTIPLICATIVE LATTICES - A GENERALIZATION OF J -IDEALS, n -IDEALS AND r -IDEALS IN RINGS SACHIN SARODE* AND VINAYAK JOSHI**
Abstract.
In this paper, we introduce a concept of X -element with respect to an M -closed set X in multiplicative lattices and study properties of X -elements. For a particular M -closed subset X , wedefine the concept of r -element, n -element and J -element. These elements generalize the notion of r -ideals, n -ideals and J -ideals of a commutative ring with unity to multiplicative lattices. In fact, weprove that an ideal I of a commutative ring R with unity is a n -ideal ( J -ideal) of R if and only if it isan n -element ( J -element) of Id ( R ) , the ideal lattice of R . Keywords:
Multiplicative lattice, prime element, X -element, n -element, J -element, r -element,commutative ring, n -ideal, r -ideal, J -ideal.1. Introduction
The ideal theory of commutative rings with unity is very rich. Many researchers defined differentideals ranging from prime ideals, maximal ideals, primary ideals to recently introduced r -ideals, n -idealsand J -ideals. More details about r -ideals, n -ideals, and J -ideals can be found in Mohamadian [11], Tekiret al. [12] and, Khashan and Bani-Ata [9] respectively.Ward and Dilworth [13] introduced the concept of multiplicative lattices to generalize the ideal theoryof commutative rings with unity. Analogously, the concepts of a prime element, maximal element,primary elements are defined.The study of prime elements and its generalization in a multiplicative lattice is the main focus ofmany researchers. Different classes of elements and generalization of a prime element in multiplicativelattices were studied; see Burton [1], Joshi and Ballal [4], Jayaram [5], Jayaram and Johnson [6, 7],Jayaram et al. [8], Manjarekar and Bingi [10].We observed a unifying pattern in the results of J -ideals, n -ideals and r -ideals of rings. This motivatesus to introduce a new class of elements, namely X -element in multiplicative lattices. Hence this studywill unify many of the results proved for these ideals and generalize it to multiplicative lattice settings.Further, for a particular M -closed subset X , we define the concept of r -element, n -element and J -element.Hence this justifies the name X -element. These elements are the generalizations of r -ideals, n -ideals and J -ideals of a commutative ring with unity. In fact, we prove that an ideal I of a commutative ring R with unity is a n -ideal( J -ideal) of R if and only if it is an n -element ( J -element) of Id ( R ) , the ideallattice of R . Date : January 17, 2021.
Mathematics Subject Classification.
Primary 13A15, 13C05, 06F10 Secondary 06A11.
Now, we begin with the necessary concepts and terminology.
Definition 1.1.
A nonempty subset I of a lattice L is a semi-ideal if x ≤ a ∈ I implies x ∈ I . Asemi-ideal I of L is an ideal if a ∨ b ∈ I whenever a, b ∈ I . An ideal (semi-ideal) I of a lattice L is a proper ideal (semi-ideal) of L if I ≠ L . A proper ideal (semi-ideal) I is prime if a ∧ b ∈ I implies a ∈ I or b ∈ I ,and it is minimal if it does not properly contain another prime ideal (prime semi-ideal). For a ∈ L , let ( a ] = { x ∈ L ∶ x ≤ a } . The set ( a ] is the principal ideal generated by a .A lattice L is complete if for any subset S of L , we have ⋁ S, ⋀ S ∈ L . The smallest element and thegreatest element of a lattice L is denoted by 0 and 1 respectively.The concept of multiplicative lattices was introduced by Ward and Dilworth [13] to study the abstractcommutative ideal theory of commutative rings.A complete lattice L is a multiplicative lattice if there exists a binary operation “ ⋅ ” called the multiplication on L satisfying the following conditions:(1) a ⋅ b = b ⋅ a , for all a, b ∈ L .(2) a ⋅ ( b ⋅ c ) = ( a ⋅ b ) ⋅ c , for all a, b, c ∈ L .(3) a ⋅ (⋁ α b α ) = ⋁ α ( a ⋅ b α ) , for all a, b α ∈ L , α ∈ Λ(an index set).(4) a ⋅ = a , for all a ∈ L .Note that in a multiplicative lattice L , a ⋅ b ≤ a ∧ b for a, b ∈ L . For this, let a = a ⋅ = a ⋅ ( b ∨ ) = a ⋅ b ∨ a .Thus a ⋅ b ≤ a . Similarly, a ⋅ b ≤ b . This proves that a ⋅ b ≤ a ∧ b . Moreover, if a ≤ b in L , then a ⋅ c ≤ b ⋅ c for every c ∈ L . Also, if a ≤ b and c ≤ d then a ⋅ c ≤ b ⋅ d .An element c of a complete lattice L is compact if c ≤ ⋁ α a α , α ∈ Λ (Λ is an index set) implies c ≤ ⋁ ni = a α i , where n ∈ Z + . The set of all compact elements of a lattice L is denoted by L ∗ .A lattice L is compactly generated or algebraic if for every x ∈ L , there exist x α ∈ L ∗ for α ∈ Λ(anindex set) such that x = ⋁ α x α , that is, every element is a join of compact elements. Equivalently, if L is a compactly generated lattice and if a /≤ b for a, b ∈ L , then there exists a nonzero compact element c ∈ L ∗ such that c ≤ a and c /≤ b .A multiplicative lattice L is 1 -compact if 1 is a compact element of L . A multiplicative lattice L is compact if every element of L is a compact element.A multiplicative lattice L is a c -lattice if L is 1-compact, compactly generated multiplicative latticein which the product of two compact elements is compact. Note that the ideal lattice of a commutativering R with unity is always a c -lattice.An element p of a multiplicative lattice L with p ≠ prime if a ⋅ b ≤ p implies a ≤ p or b ≤ p . Itis not difficult to prove that an element p (with p ≠
1) of a c -lattice L is prime if a ⋅ b ≤ p for a, b ∈ L ∗ implies a ≤ p or b ≤ p . An element p is said to be a minimal prime element if there is no prime element q such that q < p . An ideal P of a commutative ring R with unity is prime if and only if it is a primeelement of Id ( R ) , the ideal lattice of R .Let L be a c -lattice and a ∈ L . Then the radical of a is denoted by √ a and given by √ a = ⋁{ x ∈ L ∗ ∣ x n ≤ a for some n ∈ N } . Note that if any compact element c ≤ √ a , then c m ≤ a for some m ∈ N . An element a of a c -lattice is a radical element , if a = √ a . A c -lattice is called a domain if 0 is a prime element of L .A proper element i of a c -lattice L is called primary element if whenever a ⋅ b ≤ i for some a, b ∈ L then either a ≤ i or b ≤ √ i .A proper element m of a multiplicative lattice is said to be maximal , if m ≤ n <
1, then m = n .The set of all maximal elements of L is denoted by Max ( L ) . The Jacobson radical of L is the set -ELEMENTS IN MULTIPLICATIVE LATTICES 3 J ( L ) = ⋀ { m ∣ m ∈ Max ( L )} . It is easy to observe that a maximal element of a c -lattice L is prime. A c -lattice L is said to be local , if L has the unique maximal element m . In this case, we write ( L ; m ) .A non-empty subset X of L ∗ (set of all compact elements) in a c -lattice L is multiplicatively closed if s ⋅ s ∈ X , whenever s , s ∈ X .A non-empty subset X of a multiplicative lattice L is called M -closed if a, b ∈ X , then a ⋅ b ∈ X .From the definitions, it is clear that every M -closed subset A of a c -lattice is a multiplicatively closedsubset of L . The converse is not true. However, if L is a compact lattice or finite, then L = L ∗ andhence both definitions coincide with each other. Further, if p is a prime element of a c -lattice L , then L ∖ ( p ] is an M -closed subset of L .In a multiplicative lattice L , an element a ∈ L is nilpotent if a n = n ∈ Z + , and L is reduced ifthe only nilpotent element is 0. The set of all nilpotent elements of a multiplicative lattice L is denotedby Nil ( L ) .We denote the set Z ( L ) of zero-divisors in L by the set Z ( L ) = { x ∈ L ∣ x ⋅ y = y ∈ L ∖ { }} .Clearly, Nil ( L ) ⊆ Z ( L ) .Let L be a multiplicative lattice and a, b ∈ L . Then ( a ∶ b ) = ⋁ { x ∣ x ⋅ b ≤ a } . Note that x ⋅ b ≤ a ⇔ x ≤( a ∶ b ) . Clearly, a ≤ ( a ∶ b ) and ( a ∶ b ) ⋅ b ≤ a for a, b ∈ L . If a ∈ L , then ann L ( a ) = ⋁ { x ∈ L ∣ a ⋅ x = } .For undefined concepts in lattices, see Gr¨atzer [3].2. X -Elements in multiplicative Lattices We introduce the concept of an X -element in multiplicative lattices. Definition 2.1.
Let L be a multiplicative lattice and X be an M -closed subset of L . A proper element i of a multiplicative lattice L is called an X -element , if a ⋅ b ≤ i with a ∉ X implies b ≤ i for a, b ∈ L . Example 2.2.
Consider a lattice K whose Hasse diagram is shown in Figure1. On K , define the trivial multiplication x ⋅ y = = y ⋅ x for every x, y /∈ { } and x ⋅ = x = ⋅ x for every x ∈ K . It is easyto see that K is a multiplicative lattice.Moreover, K is non-reduced. If we take X = { , a, b, c, d } , then every properelement of K is an X -element of K . 0 cab d K Figure 1.
A multiplicative lattice in whichevery proper element is an X -element Remark 2.3.
Note that a proper element of a multiplicative lattice L is an X -element or not, dependson an M -closed subset X under consideration. If x is an X -element with respect to an M -closed subset X , then x may or may not be an X -element with respect to an M -closed subset X different from X .Also, note that if L is a multiplicative lattice and X = { } is an M -closed subset of L , then L doesnot contain an X -element. SACHIN SARODE* AND VINAYAK JOSHI**
Lemma 2.4.
Let L be a multiplicative lattice and X be an M -closed subset of L . If i is an X -elementof L , then ( i ] ⊆ X . In particular, if ( i ] = X , then i is an X -element of L if and only if i is a primeelement of L .Proof. Suppose i is an X -element of a multiplicative lattice L and let x ∈ ( i ] . Suppose on the contrarythat x ∉ X . Clearly, x ⋅ ≤ i with x ∉ X . Since i is an X -element, we get 1 ≤ i , a contradiction to the factthat i is a proper element of L . Therefore x ∈ X and hence ( i ] ⊆ X .Now, we prove “in particular” part. Suppose that ( i ] = X and i is an X -element of L . Let a, b ∈ L such that a ⋅ b ≤ i and a ≰ i , i.e., a ∉ X . As i is an X -element, b ≤ i . So i is a prime element of L .Conversely, suppose that ( i ] = X and i is a prime element of L . Let a, b ∈ L such that a ⋅ b ≤ i with a /∈ X = ( i ] . By primeness of i , b ≤ i . Thus i is an X -element of L . (cid:3) Remark 2.5.
The converse of the Lemma 2.4 need not be true in general, i.e., if i is a proper elementof a multiplicative lattice L such that i ∈ X , then i need not be an X -element of L . Consider the ideallattice L of the ring Z . Clearly, L is a non-reduced lattice. Put X = {( ) , ( )} . Then ( ) , ( ) ∈ X but ( ) and ( ) are not X -elements of L . Lemma 2.6.
Let L be a multiplicative lattice and X and X ′ be M -closed subsets of L such that X ⊆ X ′ .If i is an X -element of L , then i is an X ′ -element of L .Proof. follows from the definition of an X -element. (cid:3) Lemma 2.7.
Let ( L ; m ) be a local lattice. Then every proper element of L is an X -element for X = ( m ] .Proof. Let a ⋅ b ≤ i and a ∉ X = ( m ] . Since L is local, a =
1. Hence in this case b ≤ i . This proves that i is an X -element. (cid:3) Lemma 2.8.
Assume that every proper element of a c -lattice L is an X -element, where X = ( m ] and m ∈ L ∖ { } . Then m is a unique maximal element of L .Proof. Let i be a proper element of L which is an X -element. Then by Lemma 2.4, i ≤ m . This is truefor all proper elements i of L . In particular, it is true for all maximal elements m ′ too. This proves that L has unique maximal element. Therefore it is a meet-irreducible element of L . (cid:3) Lemma 2.9.
Let L be a multiplicative lattice and X be an M -closed subset of L . If { i j } , where j ∈ Λ (an index set), is a non-empty set of X -elements of L , then ⋀ j i j is also an X -element.Proof. Obvious. (cid:3)
Remark 2.10.
The join of two X -elements is not necessarily an X -element. Consider the ideal lattice L of Z with X = {( ) , ( ) , ( )} . Then ( ) , ( ) are X -elements of L , but ( ) ∨ ( ) = ( ) is not an X -element. Lemma 2.11.
Let i be a proper element of a c -lattice L and X be an M -closed subset of L . Then i isan X -element of L if and only if i = ( i ∶ a ) for all a ∉ X . In particular, if i is an X -element of L , then ( i ∶ a ) is an X -element of L for all a ∉ X .Proof. Suppose i is an X -element of L and let a ∉ X . We always have i ≤ ( i ∶ a ) . Let x be any compactelement such that x ≤ ( i ∶ a ) . Therefore x ⋅ a ≤ i . Since i is an X -element and a ∉ X , we get x ≤ i . Hence ( i ∶ a ) ≤ i , as L is a c -lattice. Therefore i = ( i ∶ a ) for all a ∉ X .Conversely, suppose that i = ( i ∶ a ) for all a ∉ X . Let c, d ∈ L such that c ⋅ d ≤ i with c ∉ X . We claimthat d ≤ i . Since c ⋅ d ≤ i , we have d ≤ ( i ∶ c ) . As c ∉ X , by the assumption ( i ∶ c ) = i , we have d ≤ i .Therefore, i is an X -element of L . Further, “in particular part” is easy to observe. (cid:3) -ELEMENTS IN MULTIPLICATIVE LATTICES 5 Lemma 2.12.
Let i be a proper element of a multiplicative lattice L and X be an M -closed subset of L . Then the following statements are equivalent.(1) i is an X -element of L .(2) ( i ∶ a ) is an X -element of L for every a ≰ i .(3) (( i ∶ a )] ⊆ X for all a ≰ i .Proof. ( ) Ô⇒ ( ) : Suppose that i is an X -element of L with j ≰ i . Clearly, ( i ∶ j ) /=
1. Let a, b ∈ L such that a ⋅ b ≤ ( i ∶ j ) with a ∉ X . So a ⋅ b ⋅ j ≤ i . As i is an X -element and a ∉ X , we get b ⋅ j ≤ i , i.e., b ≤ ( i ∶ j ) . Therefore ( i ∶ j ) is an X -element of L . ( ) Ô⇒ ( ) : follows from Lemma 2.4. ( ) Ô⇒ ( ) : Suppose that (( i ∶ a )] ⊆ X for all a ≰ i . Let c, d ∈ L such that c ⋅ d ≤ i with c ∉ X . Weclaim that d ≤ i . Suppose d ≰ i . So by the assumption and c ≤ ( i ∶ d ) , we have c ∈ X , a contradiction.Therefore d ≤ i . Hence i is an X -element of L . (cid:3) Lemma 2.13.
Let L be a multiplicative lattice and X be an M -closed subset of L . If i is a maximal X -element of L , then i is a prime element of L .Proof. Suppose i is a maximal X -element of L . Let a, b ∈ L such that a ⋅ b ≤ i and a ≰ i . Since i is an X -element and a ≰ i , by Lemma 2.12, ( i ∶ a ) is an X -element of L . As i is a maximal X -element of L and i ≤ ( i ∶ a ) , we get ( i ∶ a ) = i . Therefore b ≤ i . (cid:3) Lemma 2.14.
Let j be a proper element of a c -lattice L and X = ( j ] be an M -closed subset of L . Thena proper element i is an X -element if and only if the condition ( ∗ ) : ( ∗ ) : for all a, b ∈ L ∗ (set of all compact elements), a ⋅ b ≤ i with a ∉ X implies b ≤ i .Proof. Assume that the condition ( ∗ ) holds. Let a, b ∈ L such that a ⋅ b ≤ i with a ∉ X . As L is a c -lattice and a ≰ j , there exists ( /=) x ∈ L ∗ such that x ≤ a and x ≰ j . Now, let y be a compact elementsuch that y ≤ b . As x ⋅ y ≤ a ⋅ b ≤ i with x ∉ X = ( j ] , by the condition ( ∗ ) , y ≤ i . Thus every compactelement ≤ b is ≤ i and L is a c -lattice, we get b ≤ i . Hence i is an X -element. The converse is obvious. (cid:3) Lemma 2.15.
Let L be a c -lattice and X = ( j ] be an M -closed subset of L . Then for a prime element i of L with j ≤ i , i is an X -element if and only if i = j .Proof. Assume that i is a prime element which is also an X -element of L . By Lemma 2.4, we have i ≤ j .This together with j ≤ i , we have i = j . Conversely, assume that i = j and i is prime. To prove i is an X -element, assume that a ⋅ b ≤ i and a ∉ X . Then by primeness of i and i = j , we have b ≤ i . (cid:3) Corollary 2.16.
Let L be a c -lattice and X = ( j ] be an M -closed subset of L . Then for a maximalelement i of L , i is an X -element if and only if i = j .Proof. Assume that i is a maximal element which is also an X -element of L . By Lemma 2.4, we have i ≤ j . Thus by the maximality of i , we have i = j . Conversely, assume that i = j . Since i is maximal, itis prime. Thus the result follows from Lemma 2.15. (cid:3) Theorem 2.17.
Let L be a c -lattice and X = ( j ] be an M -closed subset of L , where j = ⋀ P where P = { i k ∣ i k is a prime elements of L } . Then the following statements are equivalent.(1) There exists an X -element in L .(2) j is a prime element of L . SACHIN SARODE* AND VINAYAK JOSHI**
Moreover, if the set
M in ( L ) of all minimal prime elements in L is finite, then all the above conditionsare equivalent to ( ∗ ) : ∣ M in ( L )∣ = .Proof. (1) ⇒ (2): Suppose there exists an X -element i in L . Let β = { x ∣ x is an X -element in L } . As i ∈ β , β is a poset under induced partial order of L . Let j ≤ j ≤ ⋯ ≤ j n ≤ ⋯ be a chain C in β . We claimthat j = ⋁ ∞ α = j α is in β , i.e., j = ⋁ ∞ α = j α is an X -element of L . Let a, b ∈ L ∗ (set of all compact elementof L ) such that a ⋅ b ≤ j = ⋁ ∞ α = j α and a /≤ j . As L is a c -lattice and a, b ∈ L ∗ , we get a ⋅ b ∈ L ∗ . Therefore a ⋅ b ≤ j ∨ j ∨ ⋯ ∨ j n for some j , j , ⋯ , j n ∈ C . Since C is a chain, we must have j ∨ j ∨ ⋯ ∨ j n = j γ forsome γ , where 1 ≤ γ ≤ n . Thus a ⋅ b ≤ j γ with a ≰ j γ . Since j γ is an X -element of L , we have b ≤ j γ . Thus b ≤ ⋁ ∞ α = j α = j . Therefore j is an X -element of L . By Zorn’s Lemma β has a maximal element w , thatis, w is a maximal X -element. By Lemma 2.13, w is a prime element of L , that is, w ∈ P . Hence j ≤ w .Also, w is an X -element, we have by Lemma 2.4, w ≤ j . Thus j = w . Hence j is prime.(2) ⇒ (1): Let j be a prime element. By Lemma 2.15, j is an X -element.We, now, prove ( ) ⇐⇒ ( ∗ ) .Assume that j is a prime element. Since j = ⋀ { i k ∣ i k is a prime elements of L } and the set M in ( L ) is finite, without loss of generality, we assume that j = n ⋀ k = { i k ∣ i k ∈ M in ( L )} . By primeness of j and j ≤ i k for all k with i k ∈ M in ( L ) , we have j = i k for all k . Thus ∣ M in ( L )∣ = ∣ M in ( L )∣ =
1. Let p be the only minimal prime element in L . Then p ≤ i k for every k . Hence j = p . This proves that j is prime. (cid:3) Lemma 2.18 (L. Fuchs and R. Reis [2, Lemma 2.5]) . Let L be a c -lattice and a ∈ L . Then radical of a is given by √ a = ⋀ { p ∈ L ∶ p is a minimal prime element over a } . Lemma 2.19.
Let L be a c -lattice and X = ( j ] be an M -closed subset of L , where j = ⋀ { i k ∣ i k is a prime element of L } . Then a proper element i is an X -element of L if and only if i isa primary element of L and √ i = j .Proof. Suppose i is an X -element of L . By Lemma 2.4, ( i ] ⊆ X . Hence i ≤ √ i ≤ √ j = j . Clearly, byLemma 2.18, j ≤ √ i . Hence j = √ i . Let a, b ∈ L such that a ⋅ b ≤ i . By Theorem 2.17, j is a primeelement of L . Hence either a ≤ j or b ≤ j . So either a ≤ √ i or b ≤ √ i . Hence i is a primary element of L . Conversely, suppose that i is a primary element of L and √ i = j . Let a, b ∈ L such that a ⋅ b ≤ i and a ≰ j . Since i is a primary element and a ≰ j = √ i , we get b ≤ i . Thus i is an X -element of L . (cid:3) Lemma 2.20.
Let L be a c -lattice and X = ( j ] be an M -closed subset of L , where j = ⋀ { i k ∣ i k is a maximal element of L } . Then a proper element i is an X -element of L if and only if i satisfies the following condition ( ∗ ) : If a ⋅ b ≤ i , then a ≤ i or b ≤ m , where m = ⋀ { i k ∣ i k is a maximal element ≥ i } and m = j .Proof. follows on similar lines as that of Lemma 2.19. (cid:3) Lemma 2.21.
Let L be a c -lattice and X be an M -closed subset of L . Let k be an element of L suchthat k ∉ X . If i and i are X -elements with i k = i k , then i = i . Further, if i is an element such that ik is an X -element, then ik = i .Proof. Clearly, i k ≤ i with k ∉ X . Since i is an X -element, we have i ≤ i . On similar lines we canprove that i ≤ i . Thus i = i . Now, we prove “further” part. Since ik is an X -element and ik ≤ ik with k ∉ X , we have i ≤ ik . The reverse inequality is always true. Hence i = ik . (cid:3) -ELEMENTS IN MULTIPLICATIVE LATTICES 7 It is well-known that if a proper ideal P of a commutative ring R with unity is prime if and only if R ∖ P is a multiplicatively closed subset of R . Analogously, a proper element p of a c -lattice L is primeif and only if L ∖ ( p ] is an M -closed subset of L . To characterize X -element, we define X -multiplicativelyclosed subset of L as follows. Definition 2.22.
Let X be an M -closed subset of a c -lattice L . A non-empty subset A of L ∗ with ( L ∗ ∖ X ) ⊆ A is called a X -multiplicatively closed subset of L , if a ∈ ( L ∗ ∖ X ) and a ∈ A , then a ⋅ a ∈ A . Remark 2.23. If A is a X -multiplicatively closed subset of L , then A need not be a multiplica-tively closed subset of L . Consider a multiplicative lattice K given in Example 2.2. If we take X = { , a, b, c, d } , then A = { , c, d } is an X -multiplicatively closed subset of K but A = { , c, d } isnot a multiplicatively closed subset of K , as c, d ∈ A and c ⋅ d = ∉ A .Also, if A is a multiplicatively closed subset of L , then A need not be a X -multiplicatively closedsubset of L for some M -closed subset X . Consider the ideal lattice L of the ring Z . Then {( )} is a multiplicatively closed subset of L , but {( )} is not a X -multiplicatively closed subset of L for X = {( ) , ( )} . Lemma 2.24.
Let i be a proper element of a c -lattice L and X be an M -closed subset of L . If i is an X -element of L , then L ∗ ∖ ( i ] is an X -multiplicatively closed subset of L . The converse is true if either X = ( j ] is an M -closed subset of a c -lattice L or L is a compact lattice.Proof. Suppose that i is an X -element of L . By Lemma 2.4, ( L ∗ ∖ X ) ⊆ ( L ∗ ∖ ( i ]) . Let a ∈ ( L ∗ ∖ X ) and b ∈ ( L ∗ ∖ ( i ]) . We claim that a ⋅ b ∈ ( L ∗ ∖ ( i ]) . Suppose on the contrary that a ⋅ b ∉ ( L ∗ ∖ ( i ]) . So a ⋅ b ≤ i . Since i is an X -element of L and a ∉ X , we get b ≤ i , a contradiction to b ∈ ( L ∗ ∖ ( i ]) . Thus a ⋅ b ∈ ( L ∗ ∖ ( i ]) . Consequently, L ∗ ∖ ( i ] is an X -multiplicatively closed subset of L .Conversely, suppose that X = ( j ] is an M -closed subset of a c -lattice L and L ∗ ∖ ( i ] is an X -multiplicatively closed subset of L . Therefore ( L ∗ ∖ ( j ]) ⊆ ( L ∗ ∖ ( i ]) . In view of Lemma 2.14,to show that i is an X -element of L , it is enough to show that for a, b ∈ L ∗ such that a ⋅ b ≤ i with a /∈ X = ( j ] , we have b ≤ i . If b ∈ ( L ∗ ∖ ( i ]) , then as ( L ∗ ∖ ( i ]) is an X -multiplicatively closed subset of L ,we get a ⋅ b ∈ ( L ∗ ∖ ( i ]) , a contradiction to a ⋅ b ≤ i . Therefore i is an X -element of L .If L is a compact lattice and X be an M -closed subset of L , then the converse follows similarly. (cid:3) Theorem 2.25.
Let j be a proper element of c -lattice L and X = ( j ] be an M -closed subset of L .Suppose a ∈ L and t ≰ a for all t ∈ A , where A is an X -multiplicatively closed subset. Then there is an X -element i of L such that a ≤ i and i is maximal with respect to t ≰ i for all t ∈ A .Proof. Let R = { c ∈ L ∣ a ≤ c and t ≰ c for all t ∈ A } . Clearly, a ∈ R and hence R is a poset underthe induced partial order of L . Let C be a chain in R and w = ⋁ { d ∣ d ∈ C } . We claim that w ∈ R .Suppose on the contrary that w ∉ R , that is, t ≤ w for some t ∈ A . Since t ∈ A ⊆ L ∗ is compact, we have t ≤ d ∨ d ∨ ⋅ ⋅ ⋅ ∨ d n for some d , d , ⋯ , d n ∈ C . As C is a chain we must have d ∨ d ∨ ⋅ ⋅ ⋅ ∨ d n = d i forsome i , where 1 ≤ i ≤ n . Thus t ≤ d i , a contradiction. Thus w ∈ R . Hence by Zorn’s lemma, there is amaximal element i of R . Hence a ≤ i and t /≤ a for all t ∈ A .In view of Lemma 2.14, to prove i is an X -element of L , assume that x, y ∈ L ∗ such that x ⋅ y ≤ i with x ∉ X = ( j ] . Suppose that y ≰ j . Clearly, y ≤ ( i ∶ x ) and, if i = ( i ∶ x ) , then y ≤ i and we are done. Henceassume that i < ( i ∶ x ) . Since i is a maximal element of R , ( i ∶ x ) /∈ R . Hence, there exists a compactelement t ∈ A such that t ≤ ( i ∶ x ) , that is, x ⋅ t ≤ i . But x ⋅ t ∈ A , as A is an X -multiplicativelyclosed subset, x ∈ ( L ∗ ∖ ( j ]) and t ∈ A . Thus there exist an element t = x ⋅ t ∈ A such that t ≤ i , acontradiction to i ∈ R . Hence i is an X -element of L . (cid:3) SACHIN SARODE* AND VINAYAK JOSHI**
Theorem 2.26.
Let L be a compact lattice and X be an M -closed subset of L . Suppose a ∈ L and t ≰ a for all t ∈ A , where A is an X -multiplicatively closed subset. Then there is an X -element i of L suchthat a ≤ i and i is maximal with respect to t ≰ i for all t ∈ A .Proof. Proof follows on similar lines as that of Theorem 2.25. (cid:3) Applications of X -Elements As already mentioned in the introduction, there is a unifying pattern in the results of J -ideals, n -idealsand r -ideals of a commutative ring with unity.In this section, we prove these results of J -ideals, n -ideals and r -ideals by suitably replacing theset X in multiplicative lattices. Hence most of the results of the papers [9], [11] and [12] becomes thecorollaries of our results.First, we quote the definitions of J -ideals, n -ideals and r -ideals using the sets Z ( R ) , N ( R ) and J ( R ) ( Z ( L ) , N ( L ) and J ( L ) ), the set of zero-divisors, the nil-radical and the Jacobson radical of acommutative ring R (multiplicative lattice L ) respectively. Definition 3.1.
A proper ideal I of a commutative ring R with unity is called: ● r -ideal, if ab ∈ I with ann R ( a ) = ( ) implies b ∈ I for all a, b ∈ R (see Mohamadian [11]). ● n -ideal, if ab ∈ I with a /∈ √ b ∈ I for all a, b ∈ R (see U. Tekir et al. [12]). ● J -ideal, if ab ∈ I with a /∈ J ( R ) implies b ∈ I for all a, b ∈ R (see Khashan and Bani-Ata [9]).Analogously, we define the concepts of r -element, n -element and J -element in multiplicative lattices. Definition 3.2.
A proper element i of a multiplicative lattice L is called: ● r -element, if a ⋅ b ≤ i with a /∈ Z ( L ) implies b ≤ i for all a, b ∈ L ∗ . ● n -element, if a ⋅ b ≤ i with a ∉ (√ ] implies b ≤ i for all a, b ∈ L ∗ . ● J -element, if a ⋅ b ≤ i with a / ∈ ( J ( L )] implies b ≤ i for all a, b ∈ L ∗ .We quote the following three results to prove that a proper ideal I of a commutative ring R withunity is an r -ideal, n -ideal and J -ideal if and only if it is an r -element, n -element and J -element of themultiplicative lattice Id ( R ) , the set of all ideals of R , respectively. Theorem 3.3 (Mohamadian [11, Lemma 2.5]) . Let R be a commutative ring with unity and I be aproper ideal of R . Then I is an r -ideal if and only if whenever J and K are ideals of R with J / ⊆ Z ( R ) and JK ⊆ I , then K ⊆ I . Theorem 3.4 (U. Tekir et al. [12, Theorem 2.7]) . Let R be a commutative ring with unity and I aproper ideal of R . Then the following are equivalent:(1) I is an n -ideal of R .(2) I = ( I ∶ a ) for every a ∉ √ .(3) For ideals J and K of R , JK ⊆ I with J ∩ ( R − √ ) /= ∅ implies K ⊆ I . Theorem 3.5 (H. A. Khashan et al. [9, Proposition 2.10]) . Let R be a commutative ring with unityand I a proper ideal of R . Then the following are equivalent:(1) I is a J -ideal of R .(2) I = ( I ∶ a ) for every a /∈ J ( R ) .(3) For ideals A and B of R , AB ⊆ I with A ⊈ J ( R ) implies B ⊆ I . -ELEMENTS IN MULTIPLICATIVE LATTICES 9 Theorem 3.6.
Let R be a Noetherian ring with unity. Then I is an r -ideal of R if and only if I is an r -element of the multiplicative lattice L = Id ( R ) , where Id ( R ) is the ideal lattice of R .Proof. Suppose that I is an r -ideal of R . Let J, K be any ideals of R such that J ⋅ K ≤ I in L with J ∉ Z ( L ) , that is, ann L ( J ) = L , where ( R ) is the least element of L , denoted by 0 L . We claim that ann R ( J ) = ( R ) . Suppose on the contrary that ( R /=) x ∈ ann R ( J ) . Hence ( x ) J = ( R ) , a contradictionto the ann L ( J ) = L . Hence ann R ( J ) = ( R ) .Now, we prove that J /⊆ Z ( R ) . Suppose on the contrary that J ⊆ Z ( R ) . Since R is Noetherian, J ⊆ Z ( R ) = ⋃ ni = P i , where P i ’s are associate primes. By Prime Avoidance Theorem J ⊆ P k for some k .Since P k is an associated prime, we have P k = ∶ x for some x ∈ R . But this will contradicts the factthat ann R ( J ) = ( ) . Hence J /⊆ Z ( R ) . By Theorem 3.3, K ⊆ I , i.e., K ≤ I . Thus I is an r -element of L .Conversely, suppose that I is a r -element of L . Let a, b ∈ R such that a ⋅ b ∈ I with ann R ( a ) = ( R ) .We claim that b ∈ I . Since ( a ⋅ b ) = ( a ) ⋅ ( b ) ⊆ I , we have a ′ ⋅ b ′ ≤ I in L , where a ′ = ( a ) , b ′ = ( b ) . Clearly, a ′ ∉ Z ( L ) . Hence b ′ ≤ I , i.e., b ∈ I . Thus I is an r -ideal of R . (cid:3) Remark 3.7.
From the proof of Theorem 3.6, it is clear that every r -element of Id ( R ) is an r -idealof R . However, for the converse we need the assumption that a ring is Noetherian. It should be notedthat if we replace “Noetherian ring” by “ring satisfies strongly annihilator condition” still the resultis true. By strong annihilator condition, we mean, for given ideal I of R , there exists a ∈ I such that ann R ( I ) = ann R ( a ) .Further, we are unable to find an example to show that the condition that the ring is Noetherian orsatisfies strongly annihilator condition is necessary to prove the above Theorem 3.6. Hence we raise thefollowing question. Question 3.8.
Let I be an r -ideal of a commutative ring R with unity. Is I an r -element of Id ( R ) ? Theorem 3.9.
Let R be a commutative ring with unity. Then I is an n -ideal of R with unity if andonly if I is a n -element of multiplicative lattice L = Id ( R ) , where Id ( R ) is the ideal lattice of R .Proof. Suppose that I is an n -ideal of R . Let J, K be any finitely generated ideals of R such that J ⋅ K ≤ I with J ≰ √ L in L . It is known that finitely generated ideals of R are compact elements of Id ( R ) . Since J ≰ √ L , we get J n /= L = ( R ) for every n ∈ N . Hence J ∩ ( R ∖ √ R ) /= ∅ . By Theorem3.4, K ⊆ I , i.e., K ≤ I . Therefore I is a n -element of L .Conversely, suppose that I is a n -element of L . Let a, b ∈ R such that a ⋅ b ∈ I with a ∉ √ R . Weclaim that b ∈ I . Since ( a ⋅ b ) = ( a ) ⋅ ( b ) ⊆ I , we have a ′ ⋅ b ′ ≤ I in L , where a ′ = ( a ) , b ′ = ( b ) ∈ L ∗ . Clearly, a ′ ≰ √ L . Hence b ′ ≤ I , i.e., b ∈ I . Thus I is an n -ideal of R . (cid:3) Theorem 3.10.
Let R be a commutative ring with unity. Then I is a J -ideal of R if and only if I isa J -element of multiplicative lattice L = Id ( R ) , where Id ( R ) is the ideal lattice of R .Proof. Suppose that I is a J -ideal of R . Let A, B be finitely generated ideals of R (which are compactelements of Id ( R ) ) such that A ⋅ B ≤ I with A ≰ J ( L ) in L = Id ( R ) . Since A ≰ J ( L ) , we get A ⊈ J ( R ) .By Theorem 3.5, B ⊆ I , i.e., B ≤ I in L . Hence I is a J -element of L .Conversely, suppose that I is a J -element of L . Let a, b ∈ R such that a ⋅ b ∈ I with a ∉ J ( R ) . Weclaim that b ∈ I . Since ( a ⋅ b ) = ( a ) ⋅ ( b ) ⊆ I , we have a ′ ⋅ b ′ ≤ I in L , where a ′ = ( a ) , b ′ = ( b ) ∈ L ∗ . Clearly, a ′ ≰ J ( L ) . Hence b ′ ≤ I , i.e., b ∈ I . Hence I is a J -ideal of R . (cid:3) Let L be a multiplicative lattice. Then one can see the each of the sets Z ( L ) , (√ ] and ( J ( L )] are multiplicatively M -closed subsets of L . So if we replace X by these sets, then we get the results of r -element, n -element and J -element respectively.We quote some of these results for ready reference.One can see that in a c -lattice L , ( L ∗ ∖ Z ( L )) ⊆ ( L ∗ ∖ ( √ ]) and ( √ ] ⊆ ( J ( L )] . For this, let x ∈ ( L ∗ ∖ Z ( L )) and x ∈ ( √ ] . Then x n = n ∈ N . Thus x ∈ Z ( L ) , a contradiction. Thisproves the inclusion ( L ∗ ∖ Z ( L )) ⊆ ( L ∗ ∖ ( √ ]) . Now, for the second inclusion, let y be any compactelement such that y ∈ ( √ ] . Then y k = k ∈ N . Let m be a maximal element of L . Then it isprime. This together with y k = ≤ m implies that y ≤ m . This further yields that y ∈ J ( L ) = ⋀ k ∈ Λ m k .Since L is a c -lattice and every compact element below √ J ( L ) , we have √ ≤ J ( L ) .Hence by Lemma 2.6, we have the following result. Proposition 3.11.
Let L be a c -lattice. Then every n -element of L is a r -element as well as it is a J -element of L . By Proposition 3.11, Theorems 3.6, 3.9, and Theorem 3.10, we have:
Proposition 3.12.
Let R be a commutative ring with unity. Then every n -ideal of R is a r -ideal aswell as it is a J -ideal. From Lemma 2.4, we get Proposition 2.2 of [9] and Proposition 2.3 of [12]. Also, Proposition 2.4of [12] follows from Lemma 2.9. It is easy to observe that Proposition 2.10 of [9] and Theorem 2.7 of[12] follows from Lemma 2.11. We observe that Proposition 2.13 of [9] follows from Lemmas 2.4 and2.13. Note that Theorem 2.17 strengthens Theorem 2.12 of [12]. Lemmas 2.19 and 2.20 generalizes theequivalence of ( i ) and ( ii ) in Corollary 2.13 of [12] and Proposition 2.20 of [9] respectively. One can seethat Lemma 2.21 extends Proposition 2.16 of [12] and Proposition 2.21 of [9]. Lastly, Theorem 2.23 of[12] and Proposition 2.29 of [9] follows from Theorem 2.25.For the following result, we need a little more explanation. Proposition 3.13 ([9, Proposition 2.3]) . Let R be a commutative ring with unity. Then the followingare equivalent. (1) R is a local ring.(2) Every proper ideal of R is a J -ideal. Proof. ( ) Ô⇒ ( ) ∶ It is clear that the ideal lattice Id ( R ) of R is a local lattice. Further, J ( L ) = m ,where m is the unique maximal element of Id ( R ) . Hence by Lemma 2.7, every proper element of L isan X -element, where X = ( m ] = ( J ( L ) ] . That is, every proper element of L is a J -element. By Theorem3.10, every proper ideal of R is a J -ideal. ( ) Ô⇒ ( ) ∶ follows from Lemma 2.8 (cid:3) Finally, the results of n -multiplicatively closed subset and J -multiplicatively closed subset can beobtained by using Lemma 2.24. Further, the results of r -ideals can be deduced from our results forNoetherian rings, since Theorem 3.6 is available for Noetherian lattice settings. If Question 3.8 has anaffirmative answer, then our results will extend most of the results of r -ideals of a commutative ringwith unity. -ELEMENTS IN MULTIPLICATIVE LATTICES 11 References [1] R. G. Burton,
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Residuated lattices , Trans. Amer. Math. Soc. (1939), 335-354.*Department of Mathematics, Shri Muktanand College, Gangapur, Dist. Aurangabad - 431 109, India. Email address : [email protected] **Department of Mathematics, Savitribai Phule Pune University, Pune-411 007, India. Email address ::