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

Emel Aslankarayigit Ugurlu
• Corresponding author
• Department of mathematics, Marmara University, Istanbul, 34722, Turkey
• Email:
/ Fethi Callialp
• Department of Mathematics, Beykent University, Istanbul, 34396, Turkey
/ Unsal Tekir
• Department of mathematics, Marmara University, Istanbul, 34722, Turkey
Published Online: 2016-09-30 | DOI: https://doi.org/10.1515/math-2016-0062

## 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.

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 1L (least element 0L) and assumed that the 1L 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 aL is said to be proper if a < 1L. If a, b belong to L, (a :Lb) is the join of all cL such that cba. 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 abe = ((a :L e) ∧ b)e for all a, bL. An element e of L is called join principal if ((aeb) :L e) = a∨(b :L e) for all a, bL. If e is meet principal and join principal, eL is said to be principal. An element p < 1L in L is said to be prime if abp implies either ap or bp for all a, bL. For any aL, he defined $\sqrt{a}$ as ∨{xL : xna for some integer n}. An element a of L is called idempotent if a2 = a. An element a of L is called compact if a ≤ ∨αi∈ Δ bαi implies abα1bα2 ∨ ... ∨ bαn such that {α1, α2, ..., αn} ⊆ 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 PGlattice (compactly generated lattice, briefly CGlattice). By a Clat tice, we mean a (not necessarily modular) complete multiplicative lattice, with the least element 0L and the compact greatest element 1L(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 < 1L in L is said to be weakly prime if 0Labp implies ap or bp for all a, bL. An element p < 1L in L is said to be almost prime if abp and abp2 imply ap or bp for all a, bL.

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 lL and BM, which satisfies the following properties for all l, lα, b in L and for all B, Bβ in M:

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

(2) (∨α lα) (∨β Bβ) = ∨α.β lαBβ;

(3) 1LB = B;

(4) 0LB = 0M.

Let M be an L— lattice module. The greatest (least) element of M is denoted by 1M (0M). An element NM is said to be proper if N < 1M . If N, K belong to M, (N :L K) is the join of all aL such that aKN. Especially, (0M :L 1M) is denoted by ann(M). In addition, if ann(M) = 0L then M is called a faithful lattice module. If aL and NM, then (N :M a) is the join of all HM such that aHN. An element N of M is called meet principal if (b ∧ (B :L N))N = bNB for all bL and for all BM. An element N of M is called join principal if b ∨ (B :L N) = ((bNB) :L N) for all bL and for all BM. 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∈Δ Bαi implies NBα1Bα2 ∨ ... ∨ Bαn for some subset {α1, α2, ..., αn} ⊆ 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 PGlattice module (compactly generated lattice, briefly CGlattice module). For various information on lattice module, one is referred to [68].

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 [1013]. 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 [1619]. 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 aL, XM such that aXN, then XN or a ≤ (N :L 1M).

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

Definition 2.2: ([8], Definition 2.1). Let M be an Llattice module and N be a proper element of M. N is called a weakly prime element of M, if whenever aL, XM such that 0MaXN, then XN or a ≤ (N :L 1M).

Definition 2.3: Let M be an Llattice module and N be a proper element of M. N is called an almost prime element of M, if whenever aL, XM such that aXN and aX ≰ (N :L 1M)N, then XN or a ≤ (N :L 1M).

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 Llattice module. The zero element 0M 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 Zmodule. Assume that (k) denotes the cyclic ideal of Z generated by kZ and < t̄ > denotes the cyclic submodule of Zmodule Z24by t̄Z24.Suppose that L = L(Z) is the set of all ideals of Z and M = L(Z24) is the set of all submodules of Zmodule Z24. There is a multiplication between elements of L and M, for every (ki) ∈ L and $<\phantom{\rule{thinmathspace}{0ex}}\overline{{t}_{j}}\phantom{\rule{thinmathspace}{0ex}}>\in \phantom{\rule{thinmathspace}{0ex}}M$ denoted by $\left({k}_{i}\right)\phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}\overline{{t}_{j}}\phantom{\rule{thinmathspace}{0ex}}>=<\overline{{k}_{i}{t}_{j}}\phantom{\rule{thinmathspace}{0ex}}>$, where ki, tjZ. Then M is a lattice module over L.Let N be the cyclic submodule of M generated by 8̄. Then clearly N = (N :L 1M)N and so N is an almost prime element. In contrast,${0}_{M}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}<\overline{0}\phantom{\rule{thinmathspace}{0ex}}>\ne \left(4\right)<\overline{4}>\le N\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}<\overline{8}\phantom{\rule{thinmathspace}{0ex}}>$ with $<\overline{4}\phantom{\rule{thinmathspace}{0ex}}>⪇\phantom{\rule{thinmathspace}{0ex}}N$ and (4) ≰ (N :L 1M) and so N is not weakly prime.

Definition 2.6: Let M be an Llattice module and N be an element of M. N is called an idempotent element of M, if N = (N :L 1M)N.

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

Definition 2.7: ([14], Definition 4). An Llattice module M is called a multiplication lattice module if for every NM, there exists aL such that N = a1M.

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

Proposition 2.8: ([14], Proposition 3). Let M be an Llattice module. Then Mis a multiplication lattice module if and only if N = (N :L 1M)1M for all NM.

We recall M/N = {BM: NB} is an L— lattice module with multiplication cD = cDN for every cL and for every NDM, [1].

Proposition 2.9: Let M be an Llattice 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 1M)N.

Proof: ⇒ Suppose N is almost prime in M. Let rL and XM/(N :L 1M)N, such that 0M/(N :L 1M)NrXN. Then we have two cases:Case 1: Suppose, on the contrary, that (N :L 1M)N = N. Then N = 0M/(N:L1M)N. Since rXN, we have NrXNrX ∨(N :L 1M)N = rXN, that is, rX = N. But then 0M/(N:L 1M)NrX = N = 0M/(N:L 1M)N, a contradiction.Case 2: Suppose that (N :L 1M)N < N. As rXN, we get rXN. Moreover, since 0M/(N :L1M)NrX = rX ∨ (N :L 1M)N, then we have rX ≰ (N :L 1M)N. Indeed, if rX ≤ (N :L 1M)N, then we get rX = rX ∨ (N :L 1M)N = (N :L 1M)N = 0M/(N :L1M)N, a contradiction. As rXN, rX ≰ (N :L 1M)N and N is almost prime in M, then we have XN or r ≤ (N :L 1M) = (N :L 1M/(N :L1M)N). Thus, N is weakly prime in M/(N :L 1M)N.⇐ Suppose N is weakly prime in M/(N :L 1M)N. Let rL and XM such that rXN and rX ≰ (N :L 1M)N. Since rX ≰ (N :L 1M)N and rX = rX ∨ (N :L 1M)N, we have rX ≠ (N :L 1M)N, i.e., 0M/(N :L1M)NrX. Moreover, as rXN then we get rXN. Since N is weakly prime in M/(N :L 1M)N, we obtain XN or r ≤ (N :L 1M/(N :L1M)N) = (N :L 1M). Thus, N is an almost prime element in M: □

Theorem 2.10: Let N be an almost prime element of an Llattice module M. If K is an element of M with KN, then N is an almost prime element of M/K.

Proof: Let rL and XM/K such that rXN and rX ≰ (N :L 1M/K) ∘ N. Firstly, we show rX ≰ (N :L 1M)N. Assume that rX ≤ (N :L 1M)N. Then we have rXK ≤ (N :L 1M)NK, i.e., rX ≤ (N :L 1M) ∘ N = (N :L 1M/K) ∘ N, which is a contradiction. Thus we get rX ≰ (N :L 1M)N. Moreover, as rXN, then we obtain rXN. Since N is an almost prime element in M, we get XN or r ≤ (N :L 1M) = (N :L 1M/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 KN. 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 Llattice module M. Then we always know that 0M/N is a weakly prime element of M/N. Hence 0M/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 Clattice Lmodule. Let N1, N2M. Suppose BM satisfies the following properties.(*) If HM is compact with HB, then either HN1 or HN2.Then either BN1 or BN2.

Proof: Assume that BN1 and BN2. Then since B is a join of compact elements, we can find compact elements H1B and H2B such that H1N1 and H2N2. Since H = H1H2B is compact, then by hypothesis (*) we have HN1 or HN2, a contradiction. Consequently, we have either BN1 or BN2. □

Theorem 3.2: Let L be a Clattice, M be a Clattice Lmodule and N be an element of M. Then the following statements are equivalent.(1) N is weakly prime in M.(2) For any aL such that a ≰ (N :L 1M), either (N :M a) = N or (N:M a) = (0M:M a).(3) For every aL* and every KM*; 0MaKN implies either a ≤ (N :L 1M) or KN.

Proof: (1) ⇒ (2) Suppose (1) holds. Let H be a compact element of M such that HB = (N :M a) and a ≰ (N :L 1M). Then aHN. We have two cases:Case 1: Let aH = 0M. Then H ≤ (0M :M a).Case 2: Let aH ≠ 0M. Since aHN, a ≰ (N :L 1M) and N is weakly prime, it follows that HN.Hence by Lemma 3.1, either (N :M a) ≤ (0M :M a) or (N :M a) ≤ N. Consequently, either (N :M a) = (0M :M a) or (N :M a) = N.(2) ⇒ (3) Suppose (2) holds. Let 0MaKN and a ≰ (N :L 1M) for aL* and KM*. We will show that KN. Since aKN, it follows that K ≤ (N :M a). If (N :M a) = N, then KN. If (N :M a) = (0M :M a), then aK = 0M. This is a contradiction. Consequently, KN.(3) ⇒ (1) Suppose (3) holds. Let aKN, a ≰ (N :L 1M) and KN for some aL and KM. Choose x1L* and Y1M* such that x1a, x1 ≰ (N :L 1M), Y1K and Y1N. Let x2a and Y2K be any two compact elements of L, M, respectively. Then by our assumption (3), we have (x2x1)(Y2Y1) = 0M and so x2Y2 = 0M. Therefore aK = 0M. This shows that N is weakly prime in M. □

Theorem 3.3: Let L be a Clattice, M be a Clattice Lmodule and N be an element of M. Then the following statements are equivalent:(1) N is almost prime in M.(2) For any aL such that a ≰ (N :L 1M), either (N :M a) = N or (N :M a) = ((N :L 1M)N :M a).(3) For every aL* and every KM*, aKN and aK ≰ (N :L 1M) N implies either a ≤ (N :L 1M) or KN.

Proof: (1) ⇒ (2) Suppose (1) holds. Let H be a compact element of M such that HB = (N :M a) and a ≰ (N :L 1M). Then aHN. We have two cases:Case 1: If aH ≤ (N :L 1M)N, then H ≤ ((N :L 1M)N:M a).Case 2: If aH ≰ (N :L 1M)N, since aHN, a ≰ (N :L 1M) and N is almost prime, it follows that HN.Hence by Lemma 3.1, we prove that either (N :M a) ≤ ((N :L 1M)N :M a) or (N :M a) ≤ N. One can see, as (N :L 1M)NN, we get ((N :L 1M)N :M a) ≤ (N:M a). Moreover, always N ≤ (N :M a). Consequently, either (N :M a) = ((N :L 1M)N:M a) or (N :M a) = N.(2) ⇒ (3) Suppose (2) holds. Let aKN and aK ≰ (N :L 1M)N for aL* and KM*. Assume that a ≰ (N :L 1M). We show that KN. Since aKN, it follows that K ≤ (N :M a). If (N :M a) = N, then KN. If (N :M a) = ((N :L 1M)N :M a), then K ≤ ((N :L 1M)N:M a). So we have aK ≤ (N :L 1M)N, a contradiction. Thus KN.(3) ⇒ (1) Suppose (3) holds. Let aKN, aK ≰ (N :L 1M)N for some aL and KM. Assume that a ≰ (N :L 1M) and KN. Choose x1L* and Y1M* such that x1a, x1 ≰ (N :L 1M), Y1K and Y1N. As L and M are C— lattices, there exist two compact elements of x2L and Y2M such that x2a and Y2K. Moreover, as x1, x2L* and Y1, Y2M*, we have x1x2L* and Y1Y2M*. Since x1a and x2a, we have x1x2a. Similarly, we have Y1Y2K. Thus (x2x1)(Y2Y1) ≤ aKN. In addition, (x2x1)(Y2Y1) ≰ (N :L 1M)N. Indeed, assume that (x2x1)(Y2Y1) ≤ (N :L 1M)N. Then we get x2Y2 ≤ (N :L 1M)N. Since x2Y2aK, we can write aK ≤ (N :L 1M)N, for x2Y2M*. But it is a contradiction.Consequently, as (x2x1)(Y2Y1) ≤ N and (x2x1)(Y2Y1) ≰ (N :L 1M)N, by our assumption (3), we have (x2x1) ≤ (N :L 1M) or (Y2Y1) ≤ N. Then we get x1 ≤ (N :L 1M) or Y1N, a contradiction. This shows that N is almost prime in M. □

Lemma 3.4: Let L be a Clattice and M be a multiplication Clattice Lmodule. If N is an almost prime element of M, then $\sqrt{\left(\left(N{:}_{L}{1}_{M}\right)N{:}_{L}{1}_{M}\right)N=\left(N{:}_{L}{1}_{M}\right)N}$.

Proof: We first note that (N :L 1M)2 ≤ ((N :L 1M)N :L 1M). Indeed, since M is a multiplication lattice module, we have (N :L 1M)(N :L 1M)1M = (N :L 1M)N, i.e., (N :L 1M)2 ≤ ((N :L 1M)N :L 1M).Let a be a compact element in L and $a\phantom{\rule{thinmathspace}{0ex}}\le \sqrt{\left(\left(N{:}_{L}{1}_{M}\right)N{:}_{L}{1}_{M}\right)}$.If a ≤ (N :L 1M), then we have a(N :L 1M) ≤ (N :L 1M)2 ≤ ((N :L 1M)N :L 1M). Thus we obtain aN = a(N :L 1M)1M ≤ (N :L 1M)N :L 1M)1M = (N :L 1M)N.If a ≰ (N :L 1M), then we have either (N :M a) = ((N :L 1M)N :M a) or (N :M a) = N by Theorem 3.3(2).Case 1: Suppose that (N :M a) = ((N :L 1M)N:M a). Since N ≤ (N:M a), then we have aNa(N :Ma) = a((N :L 1M)N :M a) ≤ (N :L 1M)N.Case 2: Suppose that (N :M a) = N. Let n be the smallest positive integer such that an ≤ ((N :L 1M)N :L1M). If n = 1, then we have a1M ≤ (N :L 1M)NN, a contradiction.So, we assume n ≥ 2. Then an1M ≤ (N :L 1M)NN with ak1M ≰ (N :L 1M)N for every kn— 1. It follows that an−11M ≤ (N :M a) = N and an−11M ≰ (N :L 1M)N. If n = 2, we also get a contradiction. If n ≥ 3, we have a(an—21M) ≤ N and a(an—2 1M) ≰ (N :L 1M)(N. Thus, since N is an almost prime element, we obtain either a ≤ (N :L 1M) or an—21MN. Continuing this process, we conclude that a ≤ (N :L 1M), which is a contradiction. Therefore $\sqrt{\left(N{:}_{L}{1}_{M}\right)N{:}_{L}{1}_{M}\right)}N\le \left(N{:}_{L}{1}_{M}\right)N$.For the second part, let a be a compact element in L and a ≤ (N :L 1M). Then we have ak 1Ma1MN for positive integer k, i.e., ak ≤ (N :L 1M). Thus, we obtain ak+1 1MakN ≤ (N :L 1M)N, i.e., ak+1 ≤ ((N :L 1M)N :L 1M). Consequently, $a\le \sqrt{\left(\left(N{:}_{L}{1}_{M}\right)N{:}_{L}{1}_{M}\right)}$

Lemma 3.5: Let L be a PG—lattice with 1L compact and M be a faithful multiplication PG—lattice module with 1M compact. Then we have (aN :L 1M) = a(N :L 1M) for every element a in L.

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

Theorem 3.6: Let L be a PG—lattice with 1L compact and M be a faithful multiplication PG—lattice L—module. For 1MNM, the followings are equivalent:(1) N is prime.(2) (N :L 1M) is prime.(3) N = q1M for 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 1L compact and M be a faithful multiplication PG—lattice module with 1M compact. For 1MNM, then the followings are equivalent:(1) N is weakly prime.(2) (N :L 1M) is weakly prime.(3) N = q1M for some weakly prime element qof L.

Proof: (1) ⇒ (2): Suppose N is weakly prime and a, bL such that 0Lab ≤ (N :L 1M). Then we have ab1MN. Since M is faithful and 0Lab, then we obtain 0Mab1M. Now, as N is weakly prime, then we get either a ≤ (N :L 1M) or b1MN (and so b ≤ (N :L 1M)). Hence (N :L 1M) is a weakly prime element in L.(2) ⇒ (1): Let (N :L 1M) be weakly prime in L. Let rL and XM, such that 0MrXN. By Lemma 3. 5, we have r(X :L 1M) = (rX :L 1M) ≤ (N :L 1M). Moreover r(X :L 1M) ≠ 0L because otherwise, if r(X :L 1M) = 0L, then rX = r(X :L 1M)1M = 0L1M = 0M. As (N :L 1M) is weakly prime, then either r ≤ (N :L 1M) or (X :L 1M) ≤ (N :L 1M). Since M is a multiplication lattice module, we obtain r ≤ (N :L 1M) or X = (X :L 1M)(1M ≤ (N :L 1M)1M = N. Thus, N is weakly prime in M.(2) ⇒ (3): Choose q = (N :L 1M).(3) ⇒ (2): Suppose that N = q1M for some weakly prime element q of L. By Lemma 3.5, we have (N :L 1M) = (q1M :L 1M) = q(1M :L 1M) = q. Thus q = (N :L 1M) is a weakly prime element. □

Theorem 3.8: Let L be a PG—lattice with 1Lcompact and M be a faithful multiplication PG—lattice module with 1M compact. For 1MNM, then the followings are equivalent:(1) N is almost prime.(2) (N :L 1M) is almost prime.(3) N = q1M for some almost prime element q of L.

Proof: (1) ⇒ (2): Suppose N is almost prime and a, bL such that ab ≤ (N :L 1M) and ab ≰ (N :L 1M)2. Then we have ab1MN and ab1M ≰ (N :L 1M)N. Indeed, if ab1M ≤ (N :L 1M)N, by Lemma 3.5, ab ≤ ((N :L 1M)N :L 1M) = (N :L 1M)(N :L 1M) = (N :L 1M)2, a contradiction. Now, N is almost prime implies that either a ≤ (N :L 1M) or b1MN (and so b ≤ (N :L 1M)). Hence (N :L 1M) is an almost prime element in L.(2) ⇒ (1): Let rL and XM such that rXN and rX ≰ (N :L 1M)N. By Lemma 3.5, we have r(X :L 1M) = (rX :L 1M) ≤ (N :L 1M). Moreover r(X :L 1M) ≰ (N :L 1M)2. Indeed, if r(X :L 1M) ≤ (N :L 1M)2 = ((N :L 1M)N :L 1M), then rX = r(X :L 1M(1M ≤ ((N :L 1M)N :L 1M)1M = (N :L1M)N, a contradiction. As (N :L 1M) is almost prime, either r ≤ (N :L 1M) or (X :L 1M) ≤ (N :L 1M). By Proposition 2.8, we have X = (X :L 1M)1M ≤ (N :L 1M)1M = N. Thus, we obtain r ≤ (N :L 1M) or XN, i.e., N is almost prime in M.(2) ⇒ (3): Choose q = (N :L 1M).(3) ⇒ (2): Suppose that N = q1M for some almost prime element q of L. By Lemma 3.5, we have (N :L 1M) = (q1M :L 1M) = q(1M :L 1M) = q. Thus q = (N :L 1M) 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 = a1M, K = b1M are two elements of M, where a, bL, the product of Nand Kis defined as NK = (a1M)(b1M) = ab1M.

Proposition 3.10: Let M be a multiplication L—lattice module and N = a1M, K = b1M are two elements of M, where a, bL. Then the product of N and K is independent of expression of Nand K.

Proof: Let N = a11M = a21M and K = b11M = b21M for a1, a2, b1, b2L. Then NK = (a1b1)1M = a1(b11M) = a1(b21M) = b2(a11M) = b2(a21M) = (a2b2)1M. □

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

Theorem 3.11: Let L be a PG—lattice with 1L compact and M be a faithful multiplication PG—lattice module with 1M 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 1M)N. As M is a multiplication lattice module, then we get N2 = NN = (N :L 1M)1M (N :L 1M)1M = (N :L 1M)21M. By Proposition 2.8 and Lemma 3.5, we obtain N = (N :L 1M)N = ((N :L 1M)N :L 1M)1M = (N :L 1M)(N :L 1M)1M = (N :L 1M)21M . Thus we have N2 = (N :L 1M)21M = N.⇐: Suppose that N2 = N. Following the same steps in the first part of the proof, we obtain N = N2 = (N :L 1M)21M = (N :L 1M)N, i.e., N = (N :L 1M)N. Consequently, N is idempotent in M. □

Theorem 3.12: Let L be a PG—lattice with 1Lcompact and M be a faithful multiplication PG—lattice module with 1M compact. Then N < 1M is prime in M if and only if whenever X and Y are elements of M such that XYN, either XNor YN.

Proof: ⇒: Assume that N is prime in M. By Theorem 3.6, we get (N :L 1M) is prime in L. Suppose that X and Y are elements of M such that XYN, but XN and YN. By Proposition 2.8, we have X = (X :L 1M)1M and Y = (Y :L 1M)1M and so XY = (X :L 1M)(Y :L 1M)1M . Since M is a multiplication lattice module, then we have (X :L 1M) ≰ (N :L 1M) and (Y :L 1M) ≰ (N :L 1M). Indeed, if (X :L 1M) ≤ (N :L 1M) and (Y :L 1M) ≤ (N :L 1M), then we have (X :L 1M)1M ≤ (N :L 1M)1M and (Y :L 1M)1M ≤ (N :L 1M)1M. So, by Proposition 2.8, XN and YN, a contradiction. Hence (X :L 1M) ≰ (N :L 1M) and (Y :L 1M) ≰ (N :L 1M). Thus, since (N :L 1M) is prime, we obtain (X :L 1M)(Y :L 1M) ≰ (N :L 1M). Moreover, we have XY = (X :L 1M)(Y :L 1M)1MN, i.e., (X :L 1M)(Y :L 1M) ≤ (N :L 1M), a contradiction. Therefore, either XN or YN.⇐: We assume that if XYN, then XN or YN. To prove that N is prime in M, it is enough, by Theorem 3.6, to prove that (N :L 1M) is prime in L. Let r1, r2L such that r1r2 ≤ (N :L 1M). Let X = r11M and Y = r21M . Then XY = r1r21MN. By assumption, either r11M = XN or r21M = YN and so, either r1 ≤ (N :L 1M) or r2 ≤ (N :L 1M). Hence (N :L 1M) 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 1L compact and M be a faithful multiplication PG—lattice module with 1M compact. Then N < 1M is weakly prime in M if and only if whenever X and Y are elements of M such that 0MXYN, either XN or YN.

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

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Accepted: 2016-08-08

Published Online: 2016-09-30

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, Export Citation