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.
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 a ∈ L is said to be proper if a < 1L. If a, b belong to L, (a :Lb) 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 < 1L 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
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 0L ≠ ab ≤ p implies a ≤ p or b ≤ p for all a, b ∈ L. An element p < 1L 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, lα, b in L and for all B, Bβ in M:
(1) (lb)B = l(bB);
(2) (∨αlα) (∨βBβ) = ∨α.βlαBβ;
(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 N ∈ M is said to be proper if N < 1M . If N, K belong to M, (N :LK) is the join of all a ∈ L such that aK ≤ N. Especially, (0M :L 1M) is denoted by ann(M). In addition, if ann(M) = 0L 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∈ΔBαi implies N ≤ Bα1 ∨ Bα2 ∨ ... ∨ 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 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
([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 1M).
Especially, M is said to be prime L— lattice module if 0M is prime element of M.
([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 0M ≠ aX ≤ N, then X ≤ N or a ≤ (N :L 1M).
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 1M)N, then X ≤ N 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:
Let M be a non-prime L— lattice 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:
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 (ki) ∈ L and
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,
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 1M)N.
Thus, any proper idempotent element of M is almost prime.
([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 = a1M.
To achieve comprehensiveness in this study, we state the following Proposition.
([14], Proposition 3). Let M be an L— lattice module. Then Mis a multiplication lattice module if and only if N = (N :L 1M)1M 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].
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 1M)N.
⇒ Suppose N is almost prime in M. Let r ∈ L and X ∈ M/(N :L 1M)N, such that 0M/(N :L 1M)N ≠ r ∘ X ≤ N. Then we have two cases:
Case 1: Suppose, on the contrary, that (N :L 1M)N = N. Then N = 0M/(N:L1M)N. Since r ∘ X ≤ N, we have N ≤ rX ∨ N ≤ rX ∨(N :L 1M)N = r ∘X ≤ N, that is, r ∘ X = N. But then 0M/(N:L 1M)N ≠ r ∘X = N = 0M/(N:L 1M)N, a contradiction.
Case 2: Suppose that (N :L 1M)N < N. As r ∘ X ≤ N, we get rX ≤ N. Moreover, since 0M/(N :L1M)N ≠ r ∘ X = rX ∨ (N :L 1M)N, then we have rX ≰ (N :L 1M)N. Indeed, if rX ≤ (N :L 1M)N, then we get r ∘ X = rX ∨ (N :L 1M)N = (N :L 1M)N = 0M/(N :L1M)N, a contradiction. As rX ≤ N, rX ≰ (N :L 1M)N and N is almost prime in M, then we have X ≤ N 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 r ∈ L and X ∈ M such that rX ≤ N and rX ≰ (N :L 1M)N. Since rX ≰ (N :L 1M)N and r ∘ X = rX ∨ (N :L 1M)N, we have r ∘ X ≠ (N :L 1M)N, i.e., 0M/(N :L1M)N ≠ r ∘ X. Moreover, as rX ≤ N then we get r ∘ X ≤ N. Since N is weakly prime in M/(N :L 1M)N, we obtain X ≤ N or r ≤ (N :L 1M/(N :L1M)N) = (N :L 1M). Thus, N is an almost prime element in M: □
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.
Let r ∈ L and X ∈ M/K such that r ∘ X ≤ N and r ∘ X ≰ (N :L 1M/K) ∘ N. Firstly, we show rX ≰ (N :L 1M)N. Assume that rX ≤ (N :L 1M)N. Then we have rX ∨ K ≤ (N :L 1M)N ∨ K, i.e., r ∘ X ≤ (N :L 1M) ∘ N = (N :L 1M/K) ∘ N, which is a contradiction. Thus we get rX ≰ (N :L 1M)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 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 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:
For any non-almost prime element N of L— lattice 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.
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.
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. □
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 1M), either (N :M a) = N or (N:M a) = (0M:M a).
(3) For every a ∈ L* and every K ∈ M*; 0M ≠ aK ≤ N implies either a ≤ (N :L 1M) or K ≤ N.
(1) ⇒ (2) Suppose (1) holds. Let H be a compact element of M such that H ≤ B = (N :M a) and a ≰ (N :L 1M). Then aH ≤ N. We have two cases:
Case 1: Let aH = 0M. Then H ≤ (0M :M a).
Case 2: Let aH ≠ 0M. Since aH ≤ N, a ≰ (N :L 1M) and N is weakly prime, it follows that H ≤ N.
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 0M ≠ aK ≤ N and a ≰ (N :L 1M) 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) = (0M :M a), then aK = 0M. This is a contradiction. Consequently, K ≤ N.
(3) ⇒ (1) Suppose (3) holds. Let aK ≤ N, a ≰ (N :L 1M) and K ≰ N for some a ∈ L and K ∈ M. Choose x1 ∈ L* and Y1 ∈ M* such that x1 ≤ a, x1 ≰ (N :L 1M), 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) = 0M and so x2Y2 = 0M. Therefore aK = 0M. This shows that N is weakly prime in M. □
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 1M), either (N :M a) = N or (N :M a) = ((N :L 1M)N :M a).
(3) For every a ∈ L* and every K ∈ M*, aK ≤ N and aK ≰ (N :L 1M) N implies either a ≤ (N :L 1M) or K ≤ N.
(1) ⇒ (2) Suppose (1) holds. Let H be a compact element of M such that H ≤ B = (N :M a) and a ≰ (N :L 1M). Then aH ≤ N. 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 aH ≤ N, a ≰ (N :L 1M) and N is almost prime, it follows that H ≤ N.
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)N ≤ N, 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 aK ≤ N and aK ≰ (N :L 1M)N for a ∈ L* and K ∈ M*. Assume that a ≰ (N :L 1M). 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 1M)N :M a), then K ≤ ((N :L 1M)N:M a). So we have aK ≤ (N :L 1M)N, a contradiction. Thus K ≤ N.
(3) ⇒ (1) Suppose (3) holds. Let aK ≤ N, aK ≰ (N :L 1M)N for some a ∈ L and K ∈ M. Assume that a ≰ (N :L 1M) and K ≰ N. Choose x1 ∈ L* and Y1 ∈ M* such that x1 ≤ a, x1 ≰ (N :L 1M), 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 1M)N. Indeed, assume that (x2 ∨ x1)(Y2 ∨ Y1) ≤ (N :L 1M)N. Then we get x2Y2 ≤ (N :L 1M)N. Since x2Y2 ≤ aK, we can write aK ≤ (N :L 1M)N, for x2Y2 ∈ M*. But it is a contradiction.
Consequently, as (x2 ∨ x1)(Y2 ∨ Y1) ≤ N and (x2 ∨ x1)(Y2 ∨ Y1) ≰ (N :L 1M)N, by our assumption (3), we have (x2 ∨ x1) ≤ (N :L 1M) or (Y2 ∨ Y1) ≤ N. Then we get x1 ≤ (N :L 1M) or Y1 ≤ N, a contradiction. This shows that N is almost prime in M. □
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
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
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 aN ≤ a(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)N ≤ N, a contradiction.
So, we assume n ≥ 2. Then an1M ≤ (N :L 1M)N ≤ N with ak1M ≰ (N :L 1M)N for every k ≤ n— 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—21M ≤ N. Continuing this process, we conclude that a ≤ (N :L 1M), which is a contradiction. Therefore
For the second part, let a be a compact element in L and a ≤ (N :L 1M). Then we have ak 1M ≤ a1M ≤ N for positive integer k, i.e., ak ≤ (N :L 1M). Thus, we obtain ak+1 1M ≤ akN ≤ (N :L 1M)N, i.e., ak+1 ≤ ((N :L 1M)N :L 1M). Consequently,
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.
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). □
Let L be a PG—lattice with 1L compact and M be a faithful multiplication PG—lattice L—module. For 1M ≠ N ∈ M, the followings are equivalent:
(1) N is prime.
(2) (N :L 1M) is prime.
(3) N = q1Mfor some prime element q of L.
The proof can be easily seen with Corollary 3 in [14]. □
Let L be a PG—lattice with 1L compact and M be a faithful multiplication PG—lattice module with 1M compact. For 1M ≠ N ∈ M, 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.
(1) ⇒ (2): Suppose N is weakly prime and a, b ∈ L such that 0L ≠ ab ≤ (N :L 1M). Then we have ab1M ≤ N. Since M is faithful and 0L ≠ ab, then we obtain 0M ≠ ab1M. Now, as N is weakly prime, then we get either a ≤ (N :L 1M) or b1M ≤ N (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 r ∈ L and X ∈ M, such that 0M ≠ rX ≤ N. 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. □
Let L be a PG—lattice with 1Lcompact and M be a faithful multiplication PG—lattice module with 1M compact. For 1M ≠ N ∈ M, 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.
(1) ⇒ (2): Suppose N is almost prime and a, b ∈ L such that ab ≤ (N :L 1M) and ab ≰ (N :L 1M)2. Then we have ab1M ≤ N 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 b1M ≤ N (and so b ≤ (N :L 1M)). Hence (N :L 1M) is an almost prime element in L.
(2) ⇒ (1): Let r ∈ L and X ∈ M such that rX ≤ N 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 X ≤ N, 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.
If Mis a multiplication L—lattice module and N = a1M, K = b1M are two elements of M, where a, b ∈ L, the product of Nand Kis defined as NK = (a1M)(b1M) = ab1M.
Let M be a multiplication L—lattice module and N = a1M, K = b1M are two elements of M, where a, b ∈ L. Then the product of N and K is independent of expression of Nand K.
Let N = a11M = a21M and K = b11M = b21M for a1, a2, b1, b2 ∈ L. 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.
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.
⇒ : 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. □
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 XY ≤ N, either X ≤ Nor Y ≤ N.
⇒: 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 XY ≤ N, but X ≰ N and Y ≰ N. 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, X ≤ N and Y ≤ N, 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)1M ≤ N, i.e., (X :L 1M)(Y :L 1M) ≤ (N :L 1M), 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 1M) is prime in L. Let r1, r2 ∈ L such that r1r2 ≤ (N :L 1M). Let X = r11M and Y = r21M . Then XY = r1r21M ≤ N. By assumption, either r11M = X ≤ N or r21M = Y ≤ N 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.
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 0M ≠ 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.
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 XY ≤ N and XY ≰ (N :L 1M)N, either X ≤ N or Y ≤ N.
References
[1] Dilworth R. P., Abstract commutative ideal theory, Pacific Journal of Mathematics, 1962, 12, 481-49810.1007/978-1-4899-3558-8_35Search in Google Scholar
[2] Anderson D. D., Multiplicative lattice, Ph. D Thesis, University of Chicago, Chicago, United States, 1974Search in Google Scholar
[3] Callialp F, Chillumuntala J., Tekir U., Weakly prime elements in multiplicative lattices, Communications in Algebra, 2012, 40: 2825-284010.1080/00927872.2011.587212Search in Google Scholar
[4] Johnson J. A., a-adic completions of Noetherian lattice modules, Fundamenta Mathematicae, 1970, 66, 341-37110.4064/fm-66-3-347-373Search in Google Scholar
[5] Johnson E. W., Johnson J. A., Lattice modules over semi-local noetherian lattice, Fundamenta Mathematicae, 1970, 68, 187-20110.4064/fm-68-2-187-201Search in Google Scholar
[6] Al-Khouja E. A., Maximal elements and prime elements in lattice modules, Damascus University for Basic Sciences, 2003, 19, 9-20Search in Google Scholar
[7] Johnson E. W., Johnson J. A., Lattice modules over element domains, Communications in Algebra, 2003, 31 (7), 3505-351810.1081/AGB-120022238Search in Google Scholar
[8] Manjarekar C. S., Kandale U. N., Weakly prime elements in lattice modules, International Journal of Scientific and Research Publications, 2013, 3(8), 1-610.1155/2014/858323Search in Google Scholar
[9] El-Bast Z. A., Smith P. F., Multiplication modules, Communications in Algebra, 1988, 16, 4, 755-77910.1080/00927878808823601Search in Google Scholar
[10] Ali M. M., Residual submodules of multiplication modules, Beitrage zur Algebra and Geometrie, 2005, 46 (2): 405-422Search in Google Scholar
[11] Ali M. M., Multiplication modules and homogeneous idealization II, Beitrage zur Algebra and Geometrie, 2007, 48(2): 321-343Search in Google Scholar
[12] Ali M. M., Smith D. J., Pure submodules of multiplication modules, Beitrage zur Algebra and Geometrie, 2004, (45), 1, 61-74Search in Google Scholar
[13] Ansari-Toroghy H., Farshadifar F., The dual notion of multiplication module, Taiwanese Journal of Mathematics, 2007, (11), 4, 1189-120110.11650/twjm/1500404812Search in Google Scholar
[14] Callialp F., Tekir U., Multiplication lattice modules, Iranian Journal of Science & Technology, 2011, 4, 309-313Search in Google Scholar
[15] Callialp F., Tekir U., Aslankarayigit E., On multiplication lattice modules, Hacettepe Journal of Mathematics and Statistics, 2014, 43 (4), 571-579Search in Google Scholar
[16] Khashan H. A., On almost prime submodules, Acta Mathematica Scientia, 2012, 32B (2): 645-65110.1016/S0252-9602(12)60045-9Search in Google Scholar
[17] Anderson D. D., Bataineh M., Generalization of prime ideals, Communications in Algebra, 2008, 36: 686-69610.1080/00927870701724177Search in Google Scholar
[18] Ansari-Toroghy H., Farshadifar F., On the dual notion of prime submodules, Algebras Colloquium, 2012, 19, (Spec 1), 1109-111610.1142/S1005386712000880Search in Google Scholar
[19] Ali M. M., Khalaf E. I., Dual notions of prime modules, Ibn al-Haitam Journal for Pure and Applied Science, 2010, 23, 226-237Search in Google Scholar
© 2016 Aslankarayigit et al., published by De Gruyter Open
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.