An Element φ-δ-Primary to another Element in Multiplicative Lattices
aa r X i v : . [ m a t h . R A ] F e b An Element φ - δ -Primary to another Element in Multiplicative LatticesA. V. Bingi Department of MathematicsSt. Xavier’s College(autonomous), Mumbai-400001, India email : ashok.bingi @ xaviers.edu Abstract
In this paper, we introduce an element φ - δ -primary to another element ina compactly generated multiplicative lattice L and obtain its characterizations.We prove many of its properties and investigate the relations between thesestructures. By a counter example, it is shown that if an element b ∈ L is φ - δ -primary to a proper element p ∈ L then b need not be δ -primary to p and foundconditions under which an element b ∈ L is δ -primary to a proper element p ∈ L if b is φ - δ -primary to p . Keywords:- expansion function, δ -primary to another element, φ - δ -primary toanother element, 2-potent δ -primary to another element, n -potent δ -primary to an-other element, idempotent element The notion of an element prime to another element in a multiplicative lattice L isintroduced by F. Alarcon et. al. in [3]. Further, the notion of an element primary toanother element in a multiplicative lattice L is introduced by C. S. Manjarekar andNitin S. Chavan in [4]. In an attempt to unify these notions of an element prime toanother element and an element primary to another element in a multiplicative lat-tice L under one frame, an element δ -primary to another element in a multiplicativelattice L is introduced by Ashok V. Bingi in [1].Further, the concept of an element weakly prime to another element and anelement weakly primary to another element in a multiplicative lattice L is introducedby C. S. Manjarekar and U. N. Kandale in [5]. To generalise these concepts, thestudy of an element φ -prime to another element and an element φ -primary to anotherelement in a multiplicative lattice L is done by Ashok V. Bingi in [2]. In this paper,we introduce and study, the notion of an element φ - δ -primary to another elementin a multiplicative lattice L as a generalization of an element δ -primary to anotherelement in L and unify an element φ -prime to another element and an element φ -primary to another element in L , under one frame.A multiplicative lattice L is a complete lattice provided with commutative, as-sociative and join distributive multiplication in which the largest element 1 acts asa multiplicative identity. An element e ∈ L is called meet principal if a ∧ be =(( a : e ) ∧ b ) e for all a, b ∈ L . An element e ∈ L is called join principal if( ae ∨ b ) : e = ( b : e ) ∨ a for all a, b ∈ L . An element e ∈ L is called principalif e is both meet principal and join principal. A multiplicative lattice L is said to beprincipally generated(PG) if every element of L is a join of principal elements of L .An element a ∈ L is called compact if for X ⊆ L , a ∨ X implies the existence ofa finite number of elements a , a , · · · , a n in X such that a a ∨ a ∨ · · · ∨ a n . Theset of compact elements of L will be denoted by L ∗ . If each element of L is a joinof compact elements of L then L is called a compactly generated lattice or simply aCG-lattice.An element a ∈ L is said to be proper if a <
1. The radical of a ∈ L is denoted by √ a and is defined as ∨{ x ∈ L ∗ | x n a , for some n ∈ Z + } . A proper element m ∈ L is said to be maximal if for every element x ∈ L such that m < x x = 1.A proper element p ∈ L is called a prime element if ab p implies a p or b p where a, b ∈ L and is called a primary element if ab p implies a p or b √ p where a, b ∈ L ∗ . For a, b ∈ L , ( a : b ) = ∨{ x ∈ L | xb a } . A multiplicative latticeis called as a Noether lattice if it is modular, principally generated and satisfiesascending chain condition. An element a ∈ L is called a zero divisor if ab = 0for some 0 = b ∈ L and is called idempotent if a = a . A multiplicative latticeis said to be a domain if it is without zero divisors and is said to be quasi-local ifit contains a unique maximal element. A quasi-local multiplicative lattice L withmaximal element m is denoted by ( L, m ). A Noether lattice L is local if it containsprecisely one maximal prime. In a Noether lattice L , an element a ∈ L is said tosatisfy restricted cancellation law if for all b, c ∈ L , ab = ac = 0 implies b = c (see[8]). According to [6], an expansion function on L is a function δ : L −→ L whichsatisfies the following two conditions: 1 ○ . a δ ( a ) for all a ∈ L , 2 ○ . a b implies δ ( a ) δ ( b ) for all a, b ∈ L . The reader is referred to [3] for general background andterminology in multiplicative lattices.According to [3], an element b ∈ L is said to be prime to a proper element p ∈ L if xb p implies x p where x ∈ L . According to [4], an element b ∈ L is saidto be primary to a proper element p ∈ L if xb p implies x √ p where x ∈ L ∗ .According to [5], an element b ∈ L is said to be weakly prime to a proper element p ∈ L if 0 = xb p implies x p where x ∈ L and an element b ∈ L is said tobe weakly primary to a proper element p ∈ L if 0 = xb p implies x √ p where x ∈ L ∗ .Further, according to [1], given an expansion function δ on L , an element b ∈ L is said to be δ -primary to a proper element p ∈ L if for all x ∈ L , xb p implies x δ ( p ). According to [2], given a function φ : L −→ L , an element b ∈ L is said tobe φ -prime to a proper element p ∈ L if for all x ∈ L , xb p and xb (cid:10) φ ( p ) implies x p and an element b ∈ L is said to be φ -primary to a proper element p ∈ L if forall x ∈ L , xb p and xb (cid:10) φ ( p ) implies x √ p .In this paper, we define an element φ - δ -primary to another element in L andobtain their characterizations. The notopn of an element φ α - δ -primary to anotherelement in L is introduced and relations among them are obtained. By counterexamples, it is shown that if b ∈ L is φ - δ -primary to a proper element of p ∈ L then b need not be φ -prime to p , b need not be prime to p and b need not be δ -primaryto p . In 6 different ways, we have proved if an element b ∈ L is φ - δ -primary toa proper element p then b is δ -primary to p under certain conditions. We definean element 2-potent δ -primary to another element of L and an element n -potent δ -primary to another element of L . Finally, we show that for an idempotent element p ∈ L , b ∈ L is φ - δ -primary to p but if b ∈ L is φ - δ -primary to a proper element p ∈ L then p need not be idempotent. Throughout this paper, 1 ○ . L denotesa compactly generated multiplicative lattice with 1 compact in which every finiteproduct of compact elements is compact, 2 ○ . δ denotes an expansion function on L and 3 ○ . φ denotes a function defined on L . φ - δ -primary to another element in L We begin with introducing the notion of an element of L to be φ - δ -primary toanother element of L which is the generalization of the concept of an element to be δ -primary to another element of L . Definition 2.1.
Given an expansion function δ : L −→ L and a function φ : L −→ L , an element b ∈ L is said to be φ - δ -primary to a proper element p ∈ L if for all x ∈ L , xb p and xb (cid:10) φ ( p ) implies x δ ( p ) . For the special functions φ α : L −→ L , an element “ φ α - δ -primary to” anotherelement in L is defined by following settings in the definition 2.1 of an element φ - δ -primary to another element in L . For any proper element p ∈ L in the definition2.1, in place of φ ( p ), set • φ ( p ) = 0. Then b ∈ L is called weakly δ -primary to p . • φ ( p ) = p . Then b ∈ L is called 2 -almost δ -primary to p or φ - δ -primaryto p or simply almost δ -primary to p . • φ n ( p ) = p n ( n > b ∈ L is called n -almost δ -primary to p or φ n - δ -primary to p ( n > • φ ω ( p ) = V ∞ i =1 p n . Then b ∈ L is called ω - δ -primary to p or φ ω - δ -primaryto p .Since for an element a ∈ L with a q but a (cid:10) φ ( q ) implies that a (cid:10) q ∧ φ ( q ),there is no loss generality in assuming that φ ( q ) q . We henceforth make thisassumption. Definition 2.2.
Given any two functions γ , γ : L −→ L , we define γ γ if γ ( a ) γ ( a ) for each a ∈ L . Clearly, we have the following order: φ φ ω · · · φ n +1 φ n · · · φ φ Further as φ ( p ) p and p δ ( p ) for each p ∈ L , the relation between thefunctions δ and φ is φ δ .According to [6], δ is an expansion function on L defined as δ ( p ) = p for each p ∈ L and δ is an expansion function on L defined as δ ( p ) = √ p for each p ∈ L .The following 2 results relate an element φ -prime to another element and anelement φ -primary to another element with some element φ - δ -primary to anotherelement in L . Theorem 2.3.
An element b ∈ L is φ - δ -primary to a proper element p ∈ L if andonly if b is φ -prime to p .Proof. The proof is obvious.
Theorem 2.4.
An element b ∈ L is φ - δ -primary to a proper element p ∈ L if andonly if b is φ -primary to p .Proof. The proof is obvious.
Theorem 2.5.
Let δ, γ : L −→ L be expansion functions on L such that δ γ . Let p ∈ L be a proper element and b ∈ L . If b is φ - δ -primary to p then b is φ - γ -primaryto p . In particular, for every expansion function δ on L , if b is φ -prime to p then b is φ - δ -primary to p .Proof. Assume that b ∈ L is φ - δ -primary to a proper element p ∈ L . Suppose xb p and xb (cid:10) φ ( p ) for x ∈ L . Then x δ ( p ) γ ( p ) and so b is φ - γ -primary to p . Next,for any expansion function δ on L , we have δ δ . So if b is φ - δ -primary to p then b is φ - δ -primary to p and we are done because if b is φ -prime to p then b is φ - δ -primary to p . Corollary 2.6.
For every expansion function δ on L , if an element b ∈ L is primeto a proper element p ∈ L then then b is φ - δ -primary to p .Proof. The proof follows by using Theorem 2.5 to the fact that if an element b ∈ L is prime to a proper element p ∈ L then then b is φ -prime to p .The following example shows that (by taking φ as φ and δ as δ for convenience) • If b ∈ L is φ - δ -primary to a proper element p ∈ L then b need not be φ -primeto p . • If b ∈ L is φ - δ -primary to a proper element p ∈ L then b need not be prime to p . Example 2.7.
Consider the lattice L of ideals of the ring R = < Z , + , · > . Thenthe only ideals of R are the principal ideals (0),(2),(3),(4),(6),(8),(12),(1). Clearly, L = { (0),(2),(3),(4),(6),(8),(12),(1) } is a compactly generated multiplicative lattice.It is easy to see that the element (2) ∈ L is φ - δ -primary to (4) ∈ L while (2) isnot φ -prime to (4) . Also (2) is not prime to (4) . Now before obtaining the characterizations of an element φ - δ -primary to anotherelement of L , we state the following essential lemma which is outcome of Lemma2.3.13 from [7]. Lemma 2.8.
Let a , a ∈ L . Suppose b ∈ L satisfies the following property:( ∗ ). If h ∈ L ∗ with h b then either h a or h a .Then either b a or b a . Theorem 2.9.
Let p be a proper element of L and b ∈ L . Then the followingstatements are equivalent: ○ . b is φ - δ -primary to p . ○ . either ( p : b ) δ ( p ) or ( p : b ) = ( φ ( p ) : b ) . ○ . for every r ∈ L ∗ , rb p and rb (cid:10) φ ( p ) implies r δ ( p ) .Proof. ○ = ⇒ ○ . Suppose 1 ○ holds. Let h ∈ L ∗ be such that h ( p : b ). Then hb p . If hb φ ( p ) then h ( φ ( p ) : b ). If hb (cid:10) φ ( p ) then since b is φ - δ -primaryto p , hb p and hb (cid:10) φ ( p ), it follows that h δ ( p ). Hence by Lemma 2 .
8, either( p : b ) ( φ ( p ) : b ) or ( p : b ) δ ( p ). Consequently, either ( p : b ) = ( φ ( p ) : b ) or( p : b ) δ ( p ).2 ○ = ⇒ ○ . Suppose 2 ○ holds. Let rb p and rb (cid:10) φ ( p ) for r ∈ L ∗ . By 2 ○ if( p : b ) = ( φ ( p ) : b ) then as r ( p : b ), it follows that r ( φ ( p ) : b ) which contradicts rb (cid:10) φ ( p ) and so we must have ( p : b ) δ ( p ). Therefore r ( p : b ) gives r δ ( p ).3 ○ = ⇒ ○ . Suppose 3 ○ holds. Let xb p and xb (cid:10) φ ( p ) for x ∈ L . Then as L is compactly generated, there exist y ′ ∈ L ∗ such that y ′ x and y ′ b (cid:10) φ ( p ). Let y x be any compact element of L . Then ( y ∨ y ′ ) ∈ L ∗ such that ( y ∨ y ′ ) b p and( y ∨ y ′ ) b (cid:10) φ ( p ). So by 3 ○ , it follows that ( y ∨ y ′ ) δ ( p ) which implies x δ ( p ) andtherefore b is φ - δ -primary to p . Theorem 2.10.
Let ( L, m ) be a quasi-local Noether lattice. If a proper element p ∈ L is such that p = m p m and b ∈ L then b is either φ - δ -primary to p or b p .Proof. Let xb p and xb (cid:10) φ ( p ) for x ∈ L . If x (cid:10) m then x = 1. So xb p gives b p . Now if x m then x m = p p and hence x δ ( p ) which implies b is φ - δ -primary to p . Thus b is either φ - δ -primary to p or b p .To obtain the relation among an element φ α - δ -primary to another element in L ,we prove the following lemma. Lemma 2.11.
Let γ , γ : L −→ L be functions on L such that γ γ and pbe proper element of L . If an element b ∈ L is γ - δ -primary to p then b ∈ L is γ - δ -primary to p.Proof. Let an element b ∈ L be γ - δ -primary to p. Suppose xb p and xb (cid:10) γ ( p )for x ∈ L . Then as γ γ , we have xb p and xb (cid:10) γ ( p ). Since b is γ - δ -primaryto p, it follows that x δ ( p ) and hence b is γ - δ -primary to p. Theorem 2.12.
For an element b ∈ L and a proper element p ∈ L , consider thefollowing statements:(a) b is δ -primary to p .(b) b is φ - δ -primary to p .(c) b is φ ω - δ -primary to p .(d) b is φ ( n +1) - δ -primary to p . (e) b is φ n - δ -primary to p where n > .(f ) b is φ - δ -primary to p .Then ( a ) = ⇒ ( b ) = ⇒ ( c ) = ⇒ ( d ) = ⇒ ( e ) = ⇒ ( f ) .Proof. Obviously, if b is δ -primary to p then b is weakly δ -primary to p and hence( a ) = ⇒ ( b ). The remaining implications follow by using Lemma 2.11 to the fact that φ φ ω · · · φ n +1 φ n · · · φ Corollary 2.13.
Let p ∈ L be a proper element and b ∈ L . Then b is φ ω - δ -primaryto p if and only if b is φ n - δ -primary to p for every n > .Proof. Assume that b is φ n - δ -primary to p for every n >
2. Let xb p and xb (cid:10) V ∞ n =1 p n for x ∈ L . Then xb p and xb (cid:10) p n for some n >
2. Since b is φ n - δ -primaryto p , we have x δ ( p ) and hence b is φ ω - δ -primary to p . The converse follows fromTheorem 2 . b ∈ L is φ n - δ -primary( n >
2) to a proper element p ∈ L then b is δ -primary to p . Theorem 2.14.
Let L be a local Noetherian domain. Let p ∈ L be a proper elementand = b ∈ L . Then b is φ n - δ -primary to p for every n > if and only if b is δ -primary to p .Proof. Assume that b is φ n - δ -primary to p for every n >
2. Let xb p for x ∈ L .If xb (cid:10) φ n ( p ) for n > b is φ n - δ -primary to p , we have x δ ( p ). If xb φ n ( p ) = p n for all n > L is local Noetherian, by Corollary 3.3 of [ ? ],it follows that xb V ∞ n =1 p n = 0 and so xb = 0. Since L is domain and 0 = b , wehave x = 0 which implies x δ ( p ). Hence, in any case, b is δ -primary to p . Conversefollows from Theorem 2.12. Corollary 2.15.
Let L be a local Noetherian domain. Let p ∈ L be a proper elementand = b ∈ L . Then b is φ ω - δ -primary to p if and only if b is δ -primary to p .Proof. The proof follows from Theorem 2 .
14 and Corollary 2 . b ∈ L is δ -primary to a proper element p ∈ L then b is φ - δ -primary to p . The following example shows that its converse is not true (bytaking φ as φ and δ as δ for convenience). Example 2.16.
Consider the lattice L of ideals of the ring R = < Z , + , · > .Then the only ideals of R are the principal ideals (0),(2),(3),(5),(6),(10),(15),(1).Clearly L = { (0),(2),(3),(5),(6),(10),(15),(1) } is a compactly generated multiplica-tive lattice. It is easy to see that the element (2) ∈ L is φ - δ -primary to (6) ∈ L but (2) is not δ -primary to (6) . In the following successive six theorems, we show conditions under which if anelement b ∈ L is φ - δ -primary to a proper element p then b is δ -primary to p . Theorem 2.17.
Let L be a Noether lattice. Let = p ∈ L be a non-nilpotent properelement satisfying the restricted cancellation law. Let b ∈ L be such that p < b .Then b is φ - δ -primary to p for some φ φ if and only if b is δ -primary to p .Proof. Assume that b ∈ L is δ -primary to p ∈ L . Then obviously, b is φ - δ -primary to p for every φ and hence for some φ φ . Conversely, assume that b is φ - δ -primaryto p for some φ φ . Then by Lemma 2.11, b is φ - δ -primary (almost δ -primary)to p . Let xb p for x ∈ L . If xb (cid:10) φ ( p ) then as b is φ - δ -primary to p , we have x δ ( p ). If xb φ ( p ) = p then xp p = 0 as p < b . Hence x p δ ( p ) byLemma 1.11 of [8] and thus b is δ -primary to p . Corollary 2.18.
Let L be a Noether lattice. Let = p ∈ L be a non-nilpotent properelement satisfying the restricted cancellation law. Let b ∈ L be such that p < b . If b is φ - δ -primary to p then b is δ -primary to p .Proof. The proof follows from proof of the Theorem 2.17.The following result is general form of Theorem 2.17.
Theorem 2.19.
Let L be a Noether lattice. Let = p ∈ L be a non-nilpotent properelement satisfying the restricted cancellation law. Let b ∈ L be such that p < b .Then b is φ - δ -primary to p for some φ φ n and for all n > if and only if b is δ -primary to p .Proof. Assume that b is δ -primary to p . Then obviously, b is φ - δ -primary to p forevery φ and hence for some φ φ n , for all n >
2. Conversely, assume that b is φ - δ -primary to p for some φ φ n and for all n >
2. Then by Lemma 2.11, b is φ n - δ -primary ( n -almost δ -primary) to p and for all n >
2. Let xb p for x ∈ L . If xb (cid:10) φ n ( p ) for some n > b is φ n - δ -primary to p , we have x δ ( p ) andwe are done. So let xb φ n ( p ) for all n >
2. Then xb p n p as n >
2. Thisimplies xp p = 0 as p < b . Hence x p δ ( p ) by Lemma 1.11 of [8] and thus b is δ -primary to p . Corollary 2.20.
Let L be a Noether lattice. Let = p ∈ L be a non-nilpotent properelement satisfying the restricted cancellation law. Let b ∈ L be such that p < b . If b is φ n - δ -primary to p ( ∀ n > then b is δ -primary to p .Proof. The proof follows from proof of the Theorem 2 . Definition 2.21.
An element b ∈ L is said to be δ -primary to a properelement p ∈ L if for all x ∈ L , xb p implies x δ ( p ) . Theorem 2.22.
Let b ∈ L be 2-potent δ -primary to a proper element p ∈ L . Then b is φ - δ -primary to p for some φ φ if and only if b is δ -primary to p .Proof. Assume that b is δ -primary to p . Then obviously, b is φ - δ -primary to p forevery φ and hence for some φ φ . Conversely, assume that b is φ - δ -primary to p for some φ φ . Then by Lemma 2.11, b is φ - δ -primary (almost δ -primary) to p .Let xb p for x ∈ L . If xb (cid:10) φ ( p ) then as b is φ - δ -primary to p , we have x δ ( p ).If xb φ ( p ) = p then as b is 2-potent δ -primary to p , we have x δ ( p ). Hence b is δ -primary to p . Corollary 2.23.
Let p ∈ L be a proper element and b ∈ L . If b is φ - δ -primary to p and b is 2-potent δ -primary to p then b is δ -primary to p .Proof. The proof follows from proof of the Theorem 2 . Definition 2.24.
Let n > . An element b ∈ L is said to be n -potent δ -primary to a proper element p ∈ L if for all x ∈ L , xb p n implies x δ ( p ) . Obviously, if an element b ∈ L is n -potent δ -primary to a proper element p ∈ L then b is 2-potent δ -primary to p .The following result is general form of Theorem 2.22. Theorem 2.25.
Let p ∈ L be a proper element and b ∈ L . Then b is φ - δ -primaryto p for some φ φ n where n > if and only if b is δ -primary to p , provided b is k -potent δ -primary to p for some k n .Proof. Assume that b is δ -primary to p . Then obviously, b is φ - δ -primary to p forevery φ and hence for some φ φ n where n >
2. Conversely, assume that b is φ - δ -primary to p for some φ φ n where n >
2. Then by Lemma 2.11, b is φ n - δ -primary ( n -almost δ -primary) to p . Let xb p for x ∈ L . If xb (cid:10) φ k ( p ) = p k then xb (cid:10) φ n ( p ) = p n as k n . Since b is φ n - δ -primary to p , we have x δ ( p ). If xb φ k ( p ) = p k then as b is k -potent δ -primary to p , we have x δ ( p ). Hence b is δ -primary to p . Corollary 2.26.
Let p ∈ L be a proper element and b ∈ L . If b is φ n - δ -primary to p and b is k -potent δ -primary to p where k n then b is δ -primary to p . Theorem 2.27.
Let p ∈ L be a proper element and b ∈ L be φ - δ -primary to p . If pb (cid:10) φ ( p ) then b is δ -primary to p .Proof. Let xb p for x ∈ L . If xb (cid:10) φ ( p ) then as b is φ - δ -primary to p , we have x δ ( p ). So assume that xb φ ( p ). Then as pb (cid:10) φ ( p ), we have db (cid:10) φ ( p ) for some d p in L . Also ( x ∨ d ) b = xb ∨ db p and ( x ∨ d ) b (cid:10) φ ( p ). As b is φ - δ -primary to p , we have x ( x ∨ d ) δ ( p ) and hence b is δ -primary to p .From the Theorem 2 .
27, it follows that, if an element b ∈ L is φ - δ -primary to aproper element p ∈ L but b is not δ -primary to p then pb φ ( p ) and hence pb p . Corollary 2.28.
If an element b ∈ L is φ - δ -primary to a proper element p ∈ L but b is not δ -primary to p then pb = 0 .Proof. The proof is obvious.
Theorem 2.29.
Let an element b ∈ L be φ - δ -primary to a proper element p ∈ L .If b is δ -primary to φ ( p ) then b is δ -primary to p .Proof. Let xb p for x ∈ L . If xb (cid:10) φ ( p ) then as b is φ - δ -primary to p , we have x δ ( p ) and we are done. Now if xb φ ( p ) then as b is δ -primary to φ ( p ), we have x δ ( φ ( p )). This implies that x δ ( p ) because φ ( p ) p and we are done.The following theorem shows that a under certain condition, b ∈ L is φ - δ -primaryto ( p : q ) ∈ L if b is φ - δ -primary to p ∈ L where q ∈ L . Theorem 2.30.
Let an element b ∈ L be φ - δ -primary to a proper element p ∈ L .Then b is φ - δ -primary to ( p : q ) for all q ∈ L if ( φ ( p ) : q ) φ ( p : q ) and ( δ ( p ) : q ) δ ( p : q ) .Proof. Let xb ( p : q ) and xb (cid:10) φ ( p : q ) for x ∈ L . Then xqb p and xqb (cid:10) φ ( p ).Now as b is φ - δ -primary to p , we have xq δ ( p ) which implies x ( δ ( p ) : q ) δ ( p : q ) and hence b is φ - δ -primary to ( p : q ). Theorem 2.31.
If an element b k ∈ L is φ - δ -primary to a proper element p ∈ L for all k ∈ Z + such that δ ( φ ( p )) = φ ( δ ( p )) then b is φ -prime to δ ( p ) where b ∈ L .Proof. Assume that xb δ ( p ) and xb (cid:10) φ ( δ ( p )) for x ∈ L . Then there exists n ∈ Z + such that x n · b n = ( xb ) n p . If ( xb ) n φ ( p ) then by hypothesis xb δ ( φ ( p )) = φ ( δ ( p )), a contradiction. So we must have x n · b n = ( xb ) n (cid:10) φ ( p ). Since b n is φ - δ -primary to p we have, x n δ ( p ) and hence x δ ( δ ( p )) = δ ( p ). Thisshows that b is φ -prime to δ ( p ).Now we relate idempotent element of L with an element φ n - δ -primary ( n > L . Theorem 2.32. If p is an idempotent element of L then b ∈ L is φ ω - δ -primary to p and hence b is φ n - δ -primary ( n > to p .Proof. As p is an idempotent element of L , we have p = p n for all n ∈ Z + . So φ ω ( p ) = p . Therefore b is φ ω - δ -primary to p . Hence b is φ n - δ -primary ( n >
2) to p ,by Theorem 2.12.As a consequence of Theorem 2.32, we have following result whose proof is ob-vious. Corollary 2.33. If p is an idempotent element of L then b ∈ L is φ - δ -primary to p . However, if b ∈ L is φ - δ -primary to p ∈ L then p need not be idempotent asshown in the following example (by taking δ as δ for convenience). Example 2.34.
Consider the lattice L of ideals of the ring R = < Z , + , · > .Then the only ideals of R are the principal ideals (0),(2),(4),(1). Clearly, L = { (0) , (2) , (4) , (1) } is a compactly generated multiplicative lattice. It is easy to seethat the element (2) ∈ L is φ - δ -primary to (4) ∈ L but (4) is not idempotent. We conclude this paper with the following examples, from which it is clear that, • If b ∈ L is φ - δ -primary to p ∈ L then b need not be 2-potent δ -primary to p . • If b ∈ L is 2-potent δ -primary to p ∈ L and b is φ - δ -primary to p then b need not be prime to p .0 Example 2.35.
Consider L as in Example . . Here the element (3) ∈ L is φ - δ -primary to (6) ∈ L but (3) is not -potent δ -primary to (6) . Example 2.36.
Consider L as in Example . . Here the element (2) ∈ L is φ - δ -primary to (4) ∈ L and (2) is -potent δ -primary to (4) but (2) is not prime to (4) . References [1] Ashok V. Bingi,
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