Prime, weakly prime and almost prime elements in multiplication lattice modules

Emel Aslankarayigit Ugurlu; Fethi Callialp; Unsal Tekir

doi:10.1515/math-2016-0062

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Prime, weakly prime and almost prime elements in multiplication lattice modules

Emel Aslankarayigit Ugurlu , Fethi Callialp und Unsal Tekir

Veröffentlicht/Copyright: 30. September 2016

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 14 Heft 1

Abstract

In this paper, we study multiplication lattice modules. We establish a new multiplication over elements of a multiplication lattice module.With this multiplication, we characterize idempotent element, prime element, weakly prime element and almost prime element in multiplication lattice modules.

Key Words: Multiplication lattice module; Prime element; Weakly prime element and Almost prime element

MSC 2010: 16F10; 16F05

1 Introduction

In 1962, R. P. Dilworth began a study of the ideal theory of commutative rings in an abstract setting in [1]. Since the investigation was to be purely ideal-theoretic, he chose to study a lattice with a commutative multiplication. Then he introduced the concept of the multiplicative lattice. By a multiplicative lat tice; R. P. Dilworth meant a complete but not necessarily modular lattice L on which there is defined a completely join distributive product. In the study, he denoted the greatest element by 1_L (least element 0_L) and assumed that the 1_L is a compact multiplicative identity. In addition, he introduced the notion of a principal element as a generalization to the notion of a principal ideal and defined the Noether lattice (see [1], Definition 3.1).

Let L be a multiplicative lattice. An element a ∈ L is said to be proper if a < 1_L. If a, b belong to L, (a :L^b) is the join of all c ∈ L such that cb ≤ a. Dilworth defined a meet (join) principal and a principal element of a multiplicative lattice as follows. An element e of L is called meet principal if a∧be = ((a :_Le) ∧ b)e for all a, b ∈ L. An element e of L is called join principal if ((ae ∨ b) :_Le) = a∨(b :_Le) for all a, b ∈ L. If e is meet principal and join principal, e ∈ L is said to be principal. An element p < 1_L in L is said to be prime if ab ≤ p implies either a ≤ p or b ≤ p for all a, b ∈ L. For any a ∈ L, he defined a as ∨{x ∈ L : xⁿ ≤ afor some integern}. An element a of L is called idempotent if a² = a. An element a of L is called compact if a ≤ ∨_{αi∈ Δ}b_αi implies a ≤ b_α1 ∨ b_α2 ∨ ... ∨ b_αn such that {α₁, α₂, ..., α_n} ⊆ Δ, where Δ is an index set. If each element of L is a join of principal (compact) elements of L, then L is called a principally generated lattice, briefly PG — lattice (compactly generated lattice, briefly CG — lattice). By a C — lat tice, we mean a (not necessarily modular) complete multiplicative lattice, with the least element 0_L and the compact greatest element 1_L(a multiplicative identity), which is generated under joins by a multiplicatively closed subset C of compact elements. For various characterizations of lattice, the reader is referred to [2].

Then in [3], F. Callialp, C. Jayaram and U. Tekir defined weakly prime and almost prime as follows: An element p < 1_L in L is said to be weakly prime if 0_L ≠ ab ≤ p implies a ≤ p or b ≤ p for all a, b ∈ L. An element p < 1_L in L is said to be almost prime if ab ≤ p and ab ≰ p² imply a ≤ p or b ≤ p for all a, b ∈ L.

In 1970, E. W. Johnson and J. A. Johnson introduced and studied Noetherian lattice modules in [4, 5]. Hence most of Dilworth’s ideas and methods were extended. Then in [2], Anderson defined lattice module as follows:

Let M be a complete lattice. Recall that M is a lat tice module over the multiplicative lattice L, or simply an L— module in case there is a multiplication between elements of L and M, denoted by lB for l ∈ L and B ∈ M, which satisfies the following properties for all l, l_α, b in L and for all B, B_β in M:

(1) (lb)B = l(bB);

(2) (∨_αl_α) (∨_βB_β) = ∨_α.βl_αB_β;

(3) 1_LB = B;

(4) 0_LB = 0_M.

Let M be an L— lattice module. The greatest (least) element of M is denoted by 1_M (0_M). An element N ∈ M is said to be proper if N < 1_M . If N, K belong to M, (N :_LK) is the join of all a ∈ L such that aK ≤ N. Especially, (0_M :_L 1_M) is denoted by ann(M). In addition, if ann(M) = 0_L then M is called a faithful lattice module. If a ∈ L and N ∈ M, then (N :_Ma) is the join of all H ∈ M such that aH ≤ N. An element N of M is called meet principal if (b ∧ (B :_LN))N = bN ∧ B for all b ∈ L and for all B ∈ M. An element N of M is called join principal if b ∨ (B :_LN) = ((bN ∨ B) :_LN) for all b ∈ L and for all B ∈ M. N is said to be principal if it is meet principal and join principal. An element N in M is called compact if N ≤ ∨_αi∈ΔB_αi implies N ≤ B_α1 ∨ B_α2 ∨ ... ∨ B_αn for some subset {α₁, α₂, ..., α_n} ⊆ Δ, where Δ is an index set. If each element of M is a join of principal (compact) elements of M, then M is called a principally generated lattice module, briefly PG — lattice module (compactly generated lattice, briefly CG — lattice module). For various information on lattice module, one is referred to [6—8].

In 1988, Z. A. El-Bast and P. F. Smith introduced the concept of multiplication module in [9]. There are many studies on multiplication modules [10—13]. With the help of the concept of multiplication module, in 2011, F. Callialp and U. Tekir defined multiplication lattice modules in [14] (see, Definition 5). They characterized multiplication lattice modules with the help of principal elements of lattice modules. In addition, they examined maximal and prime elements of lattice modules. Then in 2014, F. Callialp, U. Tekir and E. Aslankarayigit proved Nakayama Lemma for multiplication lattice modules ([15], Theorem 1. 19). Moreover in the study, the authors obtained some characterizations of maximal, prime and primary elements in multiplication lattice modules.

In this study, we continue to examine multiplication lattice modules. Our aim is to extend the concepts of almost prime ideals and idempotent ideals of commutative rings to non-modular multiplicative lattices. So, we introduce almost prime element and idempotent element in lattice modules. To define the above-mentioned elements, we use the studies [16—19]. Then we obtain the relationship between the prime (weakly prime and almost prime, respectively) element of L— module M and the prime (weakly prime and almost prime, respectively) element of L (see, Theorem3. 6-Theorem3. 8). In addition, we define a new multiplication over multiplication lattice modules (see, Definition3. 9). With the help of the multiplication, we characterize idempotent element, prime element, weakly prime element and almost prime element in Theorem3.1 1-Theorem3.1 4, respectively.

Throughout this paper, L denotes a multiplicative lattice and M denotes a complete lattice. Moreover, L_*, M_* denote the set of all compact elements of L, M, respectively.

2 Some definitions and properties

Definition 2.1

([6], Definition 3.1). Let M be an L— lattice module and N be a proper element of M. N is called a prime element of M, if whenever a ∈ L, X ∈ M such that aX ≤ N, then X ≤ N or a ≤ (N :_L 1_M).

Especially, M is said to be prime L— lattice module if 0_M is prime element of M.

Definition 2.2

([8], Definition 2.1). Let M be an L— lattice module and N be a proper element of M. N is called a weakly prime element of M, if whenever a ∈ L, X ∈ M such that 0_M ≠ aX ≤ N, then X ≤ N or a ≤ (N :_L 1_M).

Definition 2.3

Let M be an L— lattice module and N be a proper element of M. N is called an almost prime element of M, if whenever a ∈ L, X ∈ M such that aX ≤ N and aX ≰ (N :_L 1_M)N, then X ≤ N or a ≤ (N :_L 1_M).

Clearly, any prime element is weakly prime and weakly prime element is almost prime. However, any weakly prime element may not be prime, see the following example:

Example 2.4

Let M be a non-prime L— lattice module. The zero element 0_M is weakly prime, which is not prime.

For an almost prime element which is not weakly prime, we consider the following example:

Example 2.5

Let Z₂₄be Z— module. Assume that (k) denotes the cyclic ideal of Z generated by k ∈ Z and < t̄ > denotes the cyclic submodule of Z— module Z₂₄by t̄ ∈ Z₂₄.

Suppose that L = L(Z) is the set of all ideals of Z and M = L(Z₂₄) is the set of all submodules of Z— module Z₂₄. There is a multiplication between elements of L and M, for every (k_i) ∈ L and <tj¯>∈Mdenoted by(ki)<tj¯>=<kitj¯>, where k_i, t_j ∈ Z. Then M is a lattice module over L.

Let N be the cyclic submodule of M generated by 8̄. Then clearly N = (N :_L 1_M)N and so N is an almost prime element. In contrast,

0M=<0¯>≠(4)<4¯>≤N=<8¯> with <4¯>⪇Nand (4) ≰ (N :_L 1_M) and so N is not weakly prime.

Definition 2.6

Let M be an L— lattice module and N be an element of M. N is called an idempotent element of M, if N = (N :_L 1_M)N.

Thus, any proper idempotent element of M is almost prime.

Definition 2.7

([14], Definition 4). An L— lattice module M is called a multiplication lattice module if for every N ∈ M, there exists a ∈ L such that N = a1_M.

To achieve comprehensiveness in this study, we state the following Proposition.

Proposition 2.8

([14], Proposition 3). Let M be an L— lattice module. Then Mis a multiplication lattice module if and only if N = (N :_L 1_M)1_M for all N ∈ M.

We recall M/N = {B ∈ M: N ≤ B} is an L— lattice module with multiplication c ∘ D = cD ∨ N for every c ∈ L and for every N ≤ D ∈ M, [1].

Proposition 2.9

Let M be an L— lattice module and N be a proper element of M. Then N is an almost prime element in M if and only if N is a weakly prime element in M/(N :_L 1_M)N.

Proof

⇒ Suppose N is almost prime in M. Let r ∈ L and X ∈ M/(N :_L 1_M)N, such that 0_M/(N :_L 1_M)N ≠ r ∘ X ≤ N. Then we have two cases:

Case 1: Suppose, on the contrary, that (N :_L 1_M)N = N. Then N = 0_M/(N:_L1_M)N. Since r ∘ X ≤ N, we have N ≤ rX ∨ N ≤ rX ∨(N :_L 1_M)N = r ∘X ≤ N, that is, r ∘ X = N. But then 0_M/(N:_L 1_M)N ≠ r ∘X = N = 0_M/(N:_L 1_M)N, a contradiction.

Case 2: Suppose that (N :_L 1_M)N < N. As r ∘ X ≤ N, we get rX ≤ N. Moreover, since 0_M/(N :_L1_M)N ≠ r ∘ X = rX ∨ (N :_L 1_M)N, then we have rX ≰ (N :_L 1_M)N. Indeed, if rX ≤ (N :_L 1_M)N, then we get r ∘ X = rX ∨ (N :_L 1_M)N = (N :_L 1_M)N = 0_M/(N :_L1_M)N, a contradiction. As rX ≤ N, rX ≰ (N :_L 1_M)N and N is almost prime in M, then we have X ≤ N or r ≤ (N :_L 1_M) = (N :_L 1_M/(N :_L1_M)N). Thus, N is weakly prime in M/(N :_L 1_M)N.

⇐ Suppose N is weakly prime in M/(N :_L 1_M)N. Let r ∈ L and X ∈ M such that rX ≤ N and rX ≰ (N :_L 1_M)N. Since rX ≰ (N :_L 1_M)N and r ∘ X = rX ∨ (N :_L 1_M)N, we have r ∘ X ≠ (N :_L 1_M)N, i.e., 0_M/(N :_L1_M)N ≠ r ∘ X. Moreover, as rX ≤ N then we get r ∘ X ≤ N. Since N is weakly prime in M/(N :_L 1_M)N, we obtain X ≤ N or r ≤ (N :_L 1_M/(N :_L1_M)N) = (N :_L 1_M). Thus, N is an almost prime element in M: □

Theorem 2.10

Let N be an almost prime element of an L— lattice module M. If K is an element of M with K ≤ N, then N is an almost prime element of M/K.

Proof

Let r ∈ L and X ∈ M/K such that r ∘ X ≤ N and r ∘ X ≰ (N :_L 1_M/K) ∘ N. Firstly, we show rX ≰ (N :_L 1_M)N. Assume that rX ≤ (N :_L 1_M)N. Then we have rX ∨ K ≤ (N :_L 1_M)N ∨ K, i.e., r ∘ X ≤ (N :_L 1_M) ∘ N = (N :_L 1_M/K) ∘ N, which is a contradiction. Thus we get rX ≰ (N :_L 1_M)N. Moreover, as r ∘ X ≤ N, then we obtain rX ≤ N. Since N is an almost prime element in M, we get X ≤ N or r ≤ (N :_L 1_M) = (N :_L 1_M/K). Consequently, N is an almost prime element in M/K. □

Dilworth in Lemma 4.2 of [1] proved that N is a prime element of M if and only if N is a prime element of M/K, for any element K ≤ N. In the previous Theorem, we prove Lemma 4.2’s one part for an almost prime case. The other part may not be true; see the following example:

Example 2.11

For any non-almost prime element N of L— lattice module M. Then we always know that 0_M/N is a weakly prime element of M/N. Hence 0_M/N = N is a weakly prime (and so almost prime) element of M/N. However by our assumption, N is not almost prime. Consequently, N is an almost prime element of M/N, but N is not an almost prime element of M.

3 Some characterizations

In this part, we obtain several characterizations of some elements in Lattice Modules under special conditions.

Lemma 3.1

Let M be a C— lattice L— module. Let N₁, N₂ ∈ M. Suppose B ∈ M satisfies the following properties.

(*) If H ∈ M is compact with H ≤ B, then either H ≤ N₁or H ≤ N₂.

Then either B ≤ N₁or B ≤ N₂.

Proof

Assume that B ≰ N₁ and B ≰ N₂. Then since B is a join of compact elements, we can find compact elements H₁ ≤ B and H₂ ≤ B such that H₁ ≰ N₁ and H₂ ≰ N₂. Since H = H₁ ∨ H₂ ≤ B is compact, then by hypothesis (*) we have H ≤ N₁ or H ≤ N₂, a contradiction. Consequently, we have either B ≤ N₁ or B ≤ N₂. □

Theorem 3.2

Let L be a C— lattice, M be a C— lattice L— module and N be an element of M. Then the following statements are equivalent.

(1) N is weakly prime in M.

(2) For any a ∈ L such that a ≰ (N :_L 1_M), either (N :_M a) = N or (N:_M a) = (0_M:_M a).

(3) For every a ∈ L_* and every K ∈ M_*; 0_M ≠ aK ≤ N implies either a ≤ (N :_L 1_M) or K ≤ N.

Proof

(1) ⇒ (2) Suppose (1) holds. Let H be a compact element of M such that H ≤ B = (N :_M a) and a ≰ (N :_L 1_M). Then aH ≤ N. We have two cases:

Case 1: Let aH = 0_M. Then H ≤ (0_M :_M a).

Case 2: Let aH ≠ 0_M. Since aH ≤ N, a ≰ (N :_L 1_M) and N is weakly prime, it follows that H ≤ N.

Hence by Lemma 3.1, either (N :_M a) ≤ (0_M :_M a) or (N :_M a) ≤ N. Consequently, either (N :_M a) = (0_M :_M a) or (N :_M a) = N.

(2) ⇒ (3) Suppose (2) holds. Let 0_M ≠ aK ≤ N and a ≰ (N :_L 1_M) for a ∈ L_* and K ∈ M_*. We will show that K ≤ N. Since aK ≤ N, it follows that K ≤ (N :_M a). If (N :_M a) = N, then K ≤ N. If (N :_M a) = (0_M :_M a), then aK = 0_M. This is a contradiction. Consequently, K ≤ N.

(3) ⇒ (1) Suppose (3) holds. Let aK ≤ N, a ≰ (N :_L 1_M) and K ≰ N for some a ∈ L and K ∈ M. Choose x₁ ∈ L_* and Y₁ ∈ M_* such that x₁ ≤ a, x₁ ≰ (N :_L 1_M), Y₁ ≤ K and Y₁ ≰ N. Let x₂ ≤ a and Y₂ ≤ K be any two compact elements of L, M, respectively. Then by our assumption (3), we have (x₂ ∨ x₁)(Y₂ ∨ Y₁) = 0_M and so x₂Y₂ = 0_M. Therefore aK = 0_M. This shows that N is weakly prime in M. □

Theorem 3.3

Let L be a C— lattice, M be a C— lattice L— module and N be an element of M. Then the following statements are equivalent:

(1) N is almost prime in M.

(2) For any a ∈ L such that a ≰ (N :_L 1_M), either (N :_M a) = N or (N :_M a) = ((N :_L 1_M)N :_M a).

(3) For every a ∈ L_* and every K ∈ M_*, aK ≤ N and aK ≰ (N :_L 1_M) N implies either a ≤ (N :_L 1_M) or K ≤ N.

Proof

(1) ⇒ (2) Suppose (1) holds. Let H be a compact element of M such that H ≤ B = (N :_M a) and a ≰ (N :_L 1_M). Then aH ≤ N. We have two cases:

Case 1: If aH ≤ (N :_L 1_M)N, then H ≤ ((N :_L 1_M)N:_M a).

Case 2: If aH ≰ (N :_L 1_M)N, since aH ≤ N, a ≰ (N :_L 1_M) and N is almost prime, it follows that H ≤ N.

Hence by Lemma 3.1, we prove that either (N :_M a) ≤ ((N :_L 1_M)N :_M a) or (N :_M a) ≤ N. One can see, as (N :_L 1_M)N ≤ N, we get ((N :_L 1_M)N :_M a) ≤ (N:_M a). Moreover, always N ≤ (N :_M a). Consequently, either (N :_M a) = ((N :_L 1_M)N:_M a) or (N :_M a) = N.

(2) ⇒ (3) Suppose (2) holds. Let aK ≤ N and aK ≰ (N :_L 1_M)N for a ∈ L_* and K ∈ M_*. Assume that a ≰ (N :_L 1_M). We show that K ≤ N. Since aK ≤ N, it follows that K ≤ (N :_M a). If (N :_M a) = N, then K ≤ N. If (N :_M a) = ((N :_L 1_M)N :_M a), then K ≤ ((N :_L 1_M)N:_M a). So we have aK ≤ (N :_L 1_M)N, a contradiction. Thus K ≤ N.

(3) ⇒ (1) Suppose (3) holds. Let aK ≤ N, aK ≰ (N :_L 1_M)N for some a ∈ L and K ∈ M. Assume that a ≰ (N :_L 1_M) and K ≰ N. Choose x₁ ∈ L_* and Y₁ ∈ M_* such that x₁ ≤ a, x₁ ≰ (N :_L 1_M), Y₁ ≤ K and Y₁ ≰ N. As L and M are C— lattices, there exist two compact elements of x₂ ∈ L and Y₂ ∈ M such that x₂ ≤ a and Y₂ ≤ K. Moreover, as x₁, x₂ ∈ L_* and Y₁, Y₂ ∈ M_*, we have x₁ ∨ x₂ ∈ L_* and Y₁ ∨ Y₂ ∈ M_*. Since x₁ ≤ a and x₂ ≤ a, we have x₁ ∨ x₂ ≤ a. Similarly, we have Y₁ ∨ Y₂ ≤ K. Thus (x₂ ∨ x₁)(Y₂ ∨ Y₁) ≤ aK ≤ N. In addition, (x₂ ∨ x₁)(Y₂ ∨ Y₁) ≰ (N :_L 1_M)N. Indeed, assume that (x₂ ∨ x₁)(Y₂ ∨ Y₁) ≤ (N :_L 1_M)N. Then we get x₂Y₂ ≤ (N :_L 1_M)N. Since x₂Y₂ ≤ aK, we can write aK ≤ (N :_L 1_M)N, for x₂Y₂ ∈ M_*. But it is a contradiction.

Consequently, as (x₂ ∨ x₁)(Y₂ ∨ Y₁) ≤ N and (x₂ ∨ x₁)(Y₂ ∨ Y₁) ≰ (N :_L 1_M)N, by our assumption (3), we have (x₂ ∨ x₁) ≤ (N :_L 1_M) or (Y₂ ∨ Y₁) ≤ N. Then we get x₁ ≤ (N :_L 1_M) or Y₁ ≤ N, a contradiction. This shows that N is almost prime in M. □

Lemma 3.4

Let L be a C— lattice and M be a multiplication C— lattice L— module. If N is an almost prime element of M, then((N:L1M)N:L1M)N=(N:L1M)N.

Proof

We first note that (N :_L 1_M)² ≤ ((N :_L 1_M)N :_L 1_M). Indeed, since M is a multiplication lattice module, we have (N :_L 1_M)(N :_L 1_M)1_M = (N :_L 1_M)N, i.e., (N :_L 1_M)² ≤ ((N :_L 1_M)N :_L 1_M).

Let a be a compact element in L and a≤((N:L1M)N:L1M).

If a ≤ (N :_L 1_M), then we have a(N :_L 1_M) ≤ (N :_L 1_M)² ≤ ((N :_L 1_M)N :_L 1_M). Thus we obtain aN = a(N :_L 1_M)1_M ≤ (N :_L 1_M)N :_L 1_M)1_M = (N :_L 1_M)N.

If a ≰ (N :_L 1_M), then we have either (N :_M a) = ((N :_L 1_M)N :_M a) or (N :_M a) = N by Theorem 3.3(2).

Case 1: Suppose that (N :_M a) = ((N :_L 1_M)N:_M a). Since N ≤ (N:_M a), then we have aN ≤ a(N :_Ma) = a((N :_L 1_M)N :_M a) ≤ (N :_L 1_M)N.

Case 2: Suppose that (N :_M a) = N. Let n be the smallest positive integer such that aⁿ ≤ ((N :_L 1_M)N :_L1_M). If n = 1, then we have a1_M ≤ (N :_L 1_M)N ≤ N, a contradiction.

So, we assume n ≥ 2. Then aⁿ1_M ≤ (N :_L 1_M)N ≤ N with a^k1_M ≰ (N :_L 1_M)N for every k ≤ n— 1. It follows that aⁿ⁻¹1_M ≤ (N :_M a) = N and aⁿ⁻¹1_M ≰ (N :_L 1_M)N. If n = 2, we also get a contradiction. If n ≥ 3, we have a(a^n—21_M) ≤ N and a(a^n—2 1_M) ≰ (N :_L 1_M)(N. Thus, since N is an almost prime element, we obtain either a ≤ (N :_L 1_M) or a^n—21_M ≤ N. Continuing this process, we conclude that a ≤ (N :_L 1_M), which is a contradiction. Therefore (N:L1M)N:L1M)N≤(N:L1M)N.

For the second part, let a be a compact element in L and a ≤ (N :_L 1_M). Then we have a^k 1_M ≤ a1_M ≤ N for positive integer k, i.e., a^k ≤ (N :_L 1_M). Thus, we obtain a^k+1 1_M ≤ a^kN ≤ (N :_L 1_M)N, i.e., a^k+1 ≤ ((N :_L 1_M)N :_L 1_M). Consequently, a≤((N:L1M)N:L1M) □

Lemma 3.5

Let L be a PG—lattice with 1_L compact and M be a faithful multiplication PG—lattice module with 1_M compact. Then we have (aN :_L 1_M) = a(N :_L 1_M) for every element a in L.

Proof

As M is a multiplication lattice module, then we have a(N :_L 1_M)1_M = aN = (aN :_L 1_M)1_M . By Theorem 5 in [14], we obtain a(N :_L 1_M) = (aN :_L 1_M). □

Theorem 3.6

Let L be a PG—lattice with 1_L compact and M be a faithful multiplication PG—lattice L—module. For 1_M ≠ N ∈ M, the followings are equivalent:

(1) N is prime.

(2) (N :_L 1_M) is prime.

(3) N = q1_Mfor some prime element q of L.

Proof

The proof can be easily seen with Corollary 3 in [14]. □

Theorem 3.7

Let L be a PG—lattice with 1_L compact and M be a faithful multiplication PG—lattice module with 1_M compact. For 1_M ≠ N ∈ M, then the followings are equivalent:

(1) N is weakly prime.

(2) (N :_L 1_M) is weakly prime.

(3) N = q1_M for some weakly prime element qof L.

Proof

(1) ⇒ (2): Suppose N is weakly prime and a, b ∈ L such that 0_L ≠ ab ≤ (N :_L 1_M). Then we have ab1_M ≤ N. Since M is faithful and 0_L ≠ ab, then we obtain 0_M ≠ ab1_M. Now, as N is weakly prime, then we get either a ≤ (N :_L 1_M) or b1_M ≤ N (and so b ≤ (N :_L 1_M)). Hence (N :_L 1_M) is a weakly prime element in L.

(2) ⇒ (1): Let (N :_L 1_M) be weakly prime in L. Let r ∈ L and X ∈ M, such that 0_M ≠ rX ≤ N. By Lemma 3. 5, we have r(X :_L 1_M) = (rX :_L 1_M) ≤ (N :_L 1_M). Moreover r(X :_L 1_M) ≠ 0_L because otherwise, if r(X :_L 1_M) = 0_L, then rX = r(X :_L 1_M)1_M = 0_L1_M = 0_M. As (N :_L 1_M) is weakly prime, then either r ≤ (N :_L 1_M) or (X :_L 1_M) ≤ (N :_L 1_M). Since M is a multiplication lattice module, we obtain r ≤ (N :_L 1_M) or X = (X :_L 1_M)(1_M ≤ (N :_L 1_M)1_M = N. Thus, N is weakly prime in M.

(2) ⇒ (3): Choose q = (N :_L 1_M).

(3) ⇒ (2): Suppose that N = q1_M for some weakly prime element q of L. By Lemma 3.5, we have (N :_L 1_M) = (q1_M :_L 1_M) = q(1_M :_L 1_M) = q. Thus q = (N :_L 1_M) is a weakly prime element. □

Theorem 3.8

Let L be a PG—lattice with 1_Lcompact and M be a faithful multiplication PG—lattice module with 1_M compact. For 1_M ≠ N ∈ M, then the followings are equivalent:

(1) N is almost prime.

(2) (N :_L 1_M) is almost prime.

(3) N = q1_M for some almost prime element q of L.

Proof

(1) ⇒ (2): Suppose N is almost prime and a, b ∈ L such that ab ≤ (N :_L 1_M) and ab ≰ (N :_L 1_M)². Then we have ab1_M ≤ N and ab1_M ≰ (N :_L 1_M)N. Indeed, if ab1_M ≤ (N :_L 1_M)N, by Lemma 3.5, ab ≤ ((N :_L 1_M)N :_L 1_M) = (N :_L 1_M)(N :_L 1_M) = (N :_L 1_M)², a contradiction. Now, N is almost prime implies that either a ≤ (N :_L 1_M) or b1_M ≤ N (and so b ≤ (N :_L 1_M)). Hence (N :_L 1_M) is an almost prime element in L.

(2) ⇒ (1): Let r ∈ L and X ∈ M such that rX ≤ N and rX ≰ (N :_L 1_M)N. By Lemma 3.5, we have r(X :_L 1_M) = (rX :_L 1_M) ≤ (N :_L 1_M). Moreover r(X :_L 1_M) ≰ (N :_L 1_M)². Indeed, if r(X :_L 1_M) ≤ (N :_L 1_M)² = ((N :_L 1_M)N :_L 1_M), then rX = r(X :_L 1_M(1_M ≤ ((N :_L 1_M)N :_L 1_M)1_M = (N :_L1_M)N, a contradiction. As (N :_L 1_M) is almost prime, either r ≤ (N :_L 1_M) or (X :_L 1_M) ≤ (N :_L 1_M). By Proposition 2.8, we have X = (X :_L 1_M)1_M ≤ (N :_L 1_M)1_M = N. Thus, we obtain r ≤ (N :_L 1_M) or X ≤ N, i.e., N is almost prime in M.

(2) ⇒ (3): Choose q = (N :_L 1_M).

(3) ⇒ (2): Suppose that N = q1_M for some almost prime element q of L. By Lemma 3.5, we have (N :_L 1_M) = (q1_M :_L 1_M) = q(1_M :_L 1_M) = q. Thus q = (N :_L 1_M) is an almost prime element. □

Now, we define a new multiplication over the multiplication lattice modules.

Definition 3.9

If Mis a multiplication L—lattice module and N = a1_M, K = b1_M are two elements of M, where a, b ∈ L, the product of Nand Kis defined as NK = (a1_M)(b1_M) = ab1_M.

Proposition 3.10

Let M be a multiplication L—lattice module and N = a1_M, K = b1_M are two elements of M, where a, b ∈ L. Then the product of N and K is independent of expression of Nand K.

Proof

Let N = a₁1_M = a₂1_M and K = b₁1_M = b₂1_M for a₁, a₂, b₁, b₂ ∈ L. Then NK = (a₁b₁)1_M = a₁(b₁1_M) = a₁(b₂1_M) = b₂(a₁1_M) = b₂(a₂1_M) = (a₂b₂)1_M. □

With the help of the new defined multiplication, we obtain the following results.

Theorem 3.11

Let L be a PG—lattice with 1_L compact and M be a faithful multiplication PG—lattice module with 1_M compact. Then N is an idempotent element in M if and only if N² = N.

Proof

⇒ : Since N is idempotent, then we have N = (N :_L 1_M)N. As M is a multiplication lattice module, then we get N² = NN = (N :_L 1_M)1_M (N :_L 1_M)1_M = (N :_L 1_M)²1_M. By Proposition 2.8 and Lemma 3.5, we obtain N = (N :_L 1_M)N = ((N :_L 1_M)N :_L 1_M)1_M = (N :_L 1_M)(N :_L 1_M)1_M = (N :_L 1_M)²1_M . Thus we have N² = (N :_L 1_M)²1_M = N.

⇐: Suppose that N² = N. Following the same steps in the first part of the proof, we obtain N = N² = (N :_L 1_M)²1_M = (N :_L 1_M)N, i.e., N = (N :_L 1_M)N. Consequently, N is idempotent in M. □

Theorem 3.12

Let L be a PG—lattice with 1_Lcompact and M be a faithful multiplication PG—lattice module with 1_M compact. Then N < 1_M is prime in M if and only if whenever X and Y are elements of M such that XY ≤ N, either X ≤ Nor Y ≤ N.

Proof

⇒: Assume that N is prime in M. By Theorem 3.6, we get (N :_L 1_M) is prime in L. Suppose that X and Y are elements of M such that XY ≤ N, but X ≰ N and Y ≰ N. By Proposition 2.8, we have X = (X :_L 1_M)1_M and Y = (Y :_L 1_M)1_M and so XY = (X :_L 1_M)(Y :_L 1_M)1_M . Since M is a multiplication lattice module, then we have (X :_L 1_M) ≰ (N :_L 1_M) and (Y :_L 1_M) ≰ (N :_L 1_M). Indeed, if (X :_L 1_M) ≤ (N :_L 1_M) and (Y :_L 1_M) ≤ (N :_L 1_M), then we have (X :_L 1_M)1_M ≤ (N :_L 1_M)1_M and (Y :_L 1_M)1_M ≤ (N :_L 1_M)1_M. So, by Proposition 2.8, X ≤ N and Y ≤ N, a contradiction. Hence (X :_L 1_M) ≰ (N :_L 1_M) and (Y :_L 1_M) ≰ (N :_L 1_M). Thus, since (N :_L 1_M) is prime, we obtain (X :_L 1_M)(Y :_L 1_M) ≰ (N :_L 1_M). Moreover, we have XY = (X :_L 1_M)(Y :_L 1_M)1_M ≤ N, i.e., (X :_L 1_M)(Y :_L 1_M) ≤ (N :_L 1_M), a contradiction. Therefore, either X ≤ N or Y ≤ N.

⇐: We assume that if XY ≤ N, then X ≤ N or Y ≤ N. To prove that N is prime in M, it is enough, by Theorem 3.6, to prove that (N :_L 1_M) is prime in L. Let r₁, r₂ ∈ L such that r₁r₂ ≤ (N :_L 1_M). Let X = r₁1_M and Y = r₂1_M . Then XY = r₁r₂1_M ≤ N. By assumption, either r₁1_M = X ≤ N or r₂1_M = Y ≤ N and so, either r₁ ≤ (N :_L 1_M) or r₂ ≤ (N :_L 1_M). Hence (N :_L 1_M) is prime in L. Consequently, N is prime in M. □

The proof of the next Theorem can be shown to be similar to the previous proof with using Proposition 2.8 and Theorem 3.7.

Theorem 3.13

Let L be a PG—lattice with 1_L compact and M be a faithful multiplication PG—lattice module with 1_M compact. Then N < 1_M is weakly prime in M if and only if whenever X and Y are elements of M such that 0_M ≠ XY ≤ N, either X ≤ N or Y ≤ N.

Finally, the proof of the following Theorem is obtained, as in the case of Theorem 3.12, by using the proof of Proposition 2.8, Lemma 3.5 and Theorem 3.8.

Theorem 3.14

Let L be a PG—lattice with 1_L compact and M be a faithful multiplication PG—lattice module with 1_M compact. Then N < 1_M is almost prime in M if and only if whenever X and Y are elements of M such that XY ≤ N and XY ≰ (N :_L 1_M)N, either X ≤ N or Y ≤ N.

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Received: 2015-12-7

Accepted: 2016-8-8

Published Online: 2016-9-30

Published in Print: 2016-1-1

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