Some remarks on Dedekind lattices

In this paper, we prove that a principally generated C -lattice L is a Dedekind lattice if and only if L is a W I -lattice in which every invertible element is a ﬁnite meet of powers of prime elements.


Introduction
By a C-lattice L we mean a not necessarily modular complete multiplicative lattice (a(∨x i ) = ∨ax i ) generated under joins by a multiplicatively closed subset C of compact elements, with least element 0 and compact greatest element 1, operating as the multiplicative identity. In any C-lattice multiplication defines a quotient operation by a : b = ∨{x ∈ L | xb ≤ a}. Obviously C-lattices arise as abstractions of ideal systems, in particular when considering rings with identity. There the principal ideals form a generating set of compact "elements" whereas the finitely generated ideals form the set of all compact elements.
The theory of C-lattices was initiated by Dilworth in his fundamental and ground breaking paper [6] based on the notion of a principal element e. Recall that an element e ∈ L is said to be principal if it satisfies: In case that (M P) is satisfied, e is called "meet principal"; in case that (J P) is satisfied, e is called "join principal". If e satisfies (M P) only for b = 1, that is a ∧ e = (a : e)e for all a ∈ L, then e is called "weak meet principal". Finite products of meet (join) principal elements are again meet (join) principal [6,Lemmas 3.3 and 3.4]. Moreover in [2,Theorem 1.3], it is shown that principal elements are always compact. For more information on principal elements, the reader is referred to [5].
Throughout this paper L denotes a principally generated C-lattice. For the definitions of prime element, maximal element, minimal prime element, and primary element, the reader is referred to [1,7]. An element a ∈ L is called a nonzero divisor if (0 : a) = 0 and a is called invertible if a is a principal nonzero divisor. An element a ∈ L is called regular if it contains an invertible element and a is called nilpotent if a n = 0 for C. Jayaram (B) University of the West Indies, Cave Hill Campus, Bridgetown, Barbados E-mail: jayaram.chillumu@cavehill.uwi.edu some positive integer n. If 0 is the only nilpotent element, then L is called reduced. For any a, b ∈ L, we say a and b are comaximal, if a ∨ b = 1.
C-lattices can be localized. For any prime element p of L, L p denotes the localization of L at F = {x C | x p}. For details on C-lattices and their localization theory, the reader is referred to [7,12]. L is called a Prüfer lattice, if every compact element is principal. L is called a W I -lattice if every compact element a ∈ L is principal and (0 : (0 : a)) ∨ (0 : a) = 1. Note that by definition, (0 : (0 : a)).(0 : a) = 0. Prüfer lattices have been studied in [2,10]. A reduced lattice L is called quasi-regular, if for any compact element x, there is a compact element y such that (0 : (0 : x)) = (0 : y). Quasi-regular lattices have been studied in [3]. Note that by [8,Theorem 4], L is a W I -lattice if and only if L is a quasi-regular lattice whose compact elements are principal. A reduced lattice L is called a Dedekind lattice if every element not contained in any minimal prime is "weak meet principal". For various characterizations of W I -lattices and Dedekind lattices, the reader is referred to [8,9,11].
It is well known that L is a Dedekind lattice if and only if L is a W I -lattice in which every invertible element is a finite product of prime elements [11, Theorems 2.6 and 3.12]. In this paper we prove that L is a Dedekind lattice if and only if L is a W I -lattice in which every invertible element is a finite meet of powers of prime elements. For general background and terminology, the reader may consult [1,2].

Nonminimal prime elements in W I -lattices
In this section we study nonminimal prime elements in W I -lattices in which every invertible element is a finite meet of powers of prime elements. Using these results, we establish that L is a Dedekind lattice if and only if L is a W I -lattice in which every invertible element is a finite meet of powers of prime elements.
We now prove some useful lemmas. It is well known that if L is a reduced lattice, then L is a Dedekind lattice if and only if every nonminimal prime is invertible [9, Theorem 9]. The following Lemma 2.1 shows that in a W I -lattice, every nonminimal prime element is the join of invertible elements.

Lemma 2.1 Let L be a W I -lattice. Then every nonminimal prime element of L is the join of invertible elements.
Proof Let p be a nonminimal prime element of L. As L is quasi-regular, by [3, Theorem 2], there exists a compact element x ≤ p such that (0 : x) = 0. As L is a W I -lattice, x is principal, so x is invertible, and hence p is regular. Let p r = ∨{y ∈ L | y ≤ p and y is invertible}. Clearly, p r ≤ p. Suppose p r < p. Choose any principal element a ≤ p such that a ≤ p r . As L is a W I -lattice, it follows that x ∨ a is invertible, so x ∨ a ≤ p r , a contradiction. Therefore p = p r and hence every nonminimal prime element of L is the join of invertible elements. This completes the proof of the lemma.

Lemma 2.3 Let L be a W I -lattice in which every invertible element is a finite meet of powers of prime elements. Let p be a nonminimal prime which is minimal over an invertible element y ∈ L. Then p n is p-primary for all positive integers n.
Proof Let n be a positive integer and let r, s ∈ L be principal elements such that rs ≤ p n and s ≤ p. Since y n is invertible, by hypothesis, r ∨ y n = ∧ m i=1 p i α i , where p i s are prime elements of L. Since p is minimal over r ∨ y n , it follows that (r ∨ y n ) p = (rs ∨ y n ) p = ( p j α j ) p where p = p j for some j ∈ {1, 2, . . . , m}.
But ( p α j ) p ≤ ( p n ) p since rs ∨ y n ≤ p n , so α j ≥ n, therefore p α j ≤ p n and hence r ≤ p n . This shows that p n is p-primary for all positive integers n. This completes the proof of the lemma.

Lemma 2.4 Let L be a W I -lattice in which every invertible element is a finite meet of powers of prime elements. Let p be a nonidempotent, nonminimal prime element of L. Then
(i) { p n } ∞ n=1 is the set of all p-primary elements of L. (ii) p ω = ∧ ∞ n=1 p n is a prime element of L. (iii) If q < p is a prime element of L, then q ≤ p ω .
Proof (i) Note that by Lemmas 2.2 and 2.3, p n = p n+1 for all positive integers n and p n is p-primary for all positive integers n. Suppose q is p-primary. Then by [4, Lemma 3.2 (d)], q = ( p n ) p = p n , so (i) holds.
(ii) Since p p is invertible in L p , by [4, Lemma 3.2 (c)], p (ω) =∧ ∞ n=1 ( p n ) p (∧ is the meet in L p ) is a prime element of L p . It can be easily verified that p ω is a prime element of L.

Lemma 2.5 Let L be a W I -lattice in which every invertible element is a finite meet of powers of prime elements. Then every invertible element is a finite meet of primary elements.
Proof The proof of the lemma follows from Lemma 2.4 and [3, Lemma 8].
Definition 2.6 A regular prime element p of L is said to be a minimal regular prime if for any prime q < p, q is a nonregular prime element of L.

Lemma 2.7 Let L be a W I -lattice in which every invertible element is a finite meet of powers of prime elements. If p is a nonidempotent, nonminimal prime element of L, then p is a minimal regular prime element of L.
Proof Let p be a nonidempotent, nonminimal prime element of L and let q < p be a prime element of L.
Assume that q is a regular prime element of L. Suppose b ≤ q and (0 : b) = 0 for some principal element b ∈ L. Choose an invertible element a ≤ p such that p is minimal over a. Since ab is invertible, by Lemma 2.5, ab is a finite meet of primary elements of L. Let ab = ∧ n i=1 q i be a normal primary decomposition of L. Let q i ≤ p for i = 1, 2, . . . , k and q j ≤ p for j = k + 1, . . . , n. Then (ab) p = ∧ k i=1 (q i ) p . By Lemma 2.4, we can assume that √ q i ≤ p ω for i = 1, 2, . . . , k. Then a ≤ √ q i for i = 1, 2, . . . , k, so b ≤ ∧ k i=1 q i and hence a p b p = b p . Therefore, by Nakayama's lemma, b p = 0 p , a contradiction since (0 : b) = 0. This shows that p is a minimal regular prime element of L. This completes the proof of the lemma. Proof If p = p 2 , then by hypothesis, p p = y p , so by Nakayama's lemma p p = 0 p , hence y p = 0 p , a contradiction, since (0 : y) = 0. This shows that p = p 2 .

Lemma 2.9 Let L be a W I -lattice in which every invertible element is a finite meet of powers of prime elements. If p is an idempotent prime, then p is a minimal prime element of L.
Proof Suppose p is an idempotent prime element of L. Assume that p is nonminimal. Then there exists an invertible element x ≤ p. By hypothesis, x has only finitely many minimal primes, say p 1 , p 2 , . . . , p n . By Lemma 2.8, p ≤ p i for all i. As L is a W I -lattice, there exists a principal element y ≤ p such that y ≤ p i for all i. Let q ≤ p be a prime minimal over x ∨ y. If q = q 2 , then by hypothesis, (x ∨ y) q = q q , so by Nakayama's lemma q q = 0 q and therefore q is minimal, so by [3,Lemma 8], x ≤ q, a contradiction. Therefore q = q 2 and nonminimal. By hypothesis and Lemma 2.7, q is a minimal regular prime. Again since x ≤ q, it follows that p i < q for some i. This contradicts the fact that q is a minimal regular prime. Therefore p is a minimal prime element of L.

Theorem 2.10 L is a Dedekind lattice if and only if L is a W I -lattice in which every invertible element is a finite meet of powers of prime elements.
Proof If L is a Dedekind lattice, then by [11, Theorem 2.6 (viii) and Theorem 3.12], L is a W I -lattice in which every invertible element is a finite meet of powers of prime elements. Conversely, assume that L is a W I -lattice in which every invertible element is a finite meet of powers of prime elements. We claim that every invertible element is a finite product of maximal prime elements. Let a ∈ L be an invertible element and let a = ∧ n i=1 p i α i , where p i s are distinct prime elements of L. Note that by [3, Theorem 2], p i s are nonminimal prime elements of L. Again by Lemmas 2.7 and 2.9, each p i is maximal and so they are pairwise comaximal. Consequently, a is a finite product of maximal prime elements. Now the result follows from [11, Theorems 2.6 (viii) and 3.12]. This completes the proof of the theorem.
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