Classical Quasi Primary Elements in Lattice Modules

Transcription

1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 2, 2015, ISSN Classical Quasi Primary Elements in Lattice Modules C. S. Manjarekar 1, U. N. Kandale 2, 1 Department of Mathematics, Shivaji University, Kolhapur, India 2 Department of General Engineering, Sharad Institute, Shivaji University, Kolhapur, India Abstract. In this paper we introduce the notion of classical primary and classical quasi primary elements in lattice modules which are the generalization of the concepts in submodules. We obtain some characterizations of classical primary and classical quasi primary elements.we also investigate the decomposition and minimal decomposition into classical quasi primary elements Mathematics Subject Classifications: 13A99 Key Words and Phrases: Primary element, prime element,classical quasi primary element,lattice modules. 1. Introduction Multiplicative lattice L is a complete lattice provided with commutative,associative and join distributive multiplication for which the largest element 1 acts as an multiplicative identity. A proper element p of L is called prime element if abpimplies apor bpfor a, b L and is called primary element if abpimplies apor b n p for some positive integer n. For a L, a= {x L x n a for some integer n}. Let L be a multiplicative lattice. A lattice module over L or simply a lattice module is defined to be a complete lattice M with multiplication L M M satisfying, (i) ( α a α )A= α a α A a α L, A M for some integer (ii) a( α A α )= α aa α a L, A α M (iii) (ab)a= a(ba) a, b L, A M (iv) IA= A A M (v) OA= O M A M where O M = gl b(m). Corresponding author. addresses: csmanjrekar@yahoo.co.in (C. Manjarekar),ujwalabiraje@gmail.com (U. Kandale) c 2015 EJPAM All rights reserved.

2 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), Elements of L will generally be denoted by a, b, c,... except that the least element of L will be denoted by O and greatest element of L will be denoted by 1. The elements of M will generally be denoted by A, B, C,... except that the least element and the greatest element of M will be denoted by O M and I M. Here after L will be a multiplicative lattice and M will be a lattice module over L. As in the case of commutative rings, there are residuation operations in lattice module. For a, b L and A, B M, a : b is the join of all elements c in L such that cba, A : b is the join of all elements C in M such that bc A and A : B is the join of all elements a in L such that ab A. An element N I M of a lattice module M is called a prime element if whenever aa N where a L,A M implies either a(n : I M ) or A N. An element N I M of a lattice module M is called a primary element if whenever aa N where a L,A M implies either A N or a n (N : I M ) for some positive integer n. A lattice module M is called a multiplication lattice module if for any element N of M there exists an element a of L such that N= ai M.An element N I M of a lattice module M is said to have primary decomposition if there exist primary elements Q 1,Q 2,...,Q k such that N= Q 1 Q 2 Q k. If some Q i contains the meet of remaining ones then this Q i can be dropped from the primary decomposition. Similarly any other primary component which contains the meet of remaining ones can be dropped from the primary decomposition. If such primary components are removed and the primary components with same associated primes are combined then we get a reduced primary decomposition in which distinct primary components are associated with distinct primes such a primary decomposition is called a normal decomposition. This decomposition is also said to be reduced. This study is carried out by D. D. Anderson[1] and for multiplicative lattices this work is done by R. P. Dilworth[2]. 2. Classical Primary and Classical Quasi Primary Elements The notion of a quasi primary ideal was defined by Fuchs[4] which is a generalization of the notions of a primary ideal. The notions of quasi primary, classical primary, classical quasi primary sub modules are studied by M Behboodi et al. [1]. We generalize there notions for multiplicative lattices and lattice modules. Definition 1. An element q of a multiplicative lattice L is called a classical primary element if abr q where a, b L, r Limplies that either ar q or b k r q for some integer k. Definition 2. An element q of a multiplicative lattice L is called a classical quasi primary element if abr q where a, b L, r Limplies that either a k r q or b k r q for some integer k. Definition 3. An element q of a multiplicative lattice L is called a quasi primary element if radical of q is a prime element that is q is called quasi primary if ab q where a, b L implies that either a k q or b k q for some integer k.

3 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), Definition 4. Let M be a lattice module over a multiplicative lattice L A proper element Q of M is called a classical primary element in M if abn Q where a, b L, N M then either an Q or b k N Q for some integer k. Definition 5. A proper element Q of M is called a classical quasi primary element in M if abn Q where a, b L, N M then either a k N Q or b k N Q for some integer k. Definition 6. A proper element Q of M is called a quasi primary element if (Q : I M ) is a prime element of L. Example 1. Let R be a integral domain and L(R) denote the set of all ideals of R. Then L(R) is a multiplicative lattice. Let F= λ R λ be a free R-module and let M=L(F) denote the set of all submodules of F. Then M is a lattice module over a multiplicative lattice L(R). Assume that, P is a nonzero prime ideal in R. Let N= A be a proper submodule of F such that for every either A = P or A =(0). Then N is a classical primary element of M. It can be verified that, if there exist 1, 2 such that A 1 = P and A 2 =(0) then N is not a primary element of M, see[1]. Example 2. Let L(Z) denote the set of all ideals of Z, the set of integers. If p is a prime integer then Z(p )={ a p k + Z a, k are integers andk Z + } is a module over Z. Let M=L(Z(p )) denote the set of all submodules of Z(p ). Then M is a lattice module over a multiplicative lattice L(Z). Every nonzero proper submodule of Z(p ) is a classical primary but not a primary element of M, see[1]. Example 3. Let R= Z and M = Q where Q is a module over R= Z. Let L(R) denote the set of all ideals of Z and L(Q) denote the set of submodules of Q. Then L(Q) is a lattice module over L(R). Each proper submodule N of M is a quasi primary element since, (N : Q)=(0). If N= Z+ Z.( 1 5 ), the submodule of M generated by{1, }, then N, but for each k 1, 2 k N and 3k N. Thus, N is not a classical quasi primary element of L(Q) see[1]. Example 4. Let R= Z, M= Z Z where M is a module over R= Z.Let L(R) denote the set of all ideals of R and L(M) denote the set of all submodules of M. Then L(M) is a lattice module over L(R). Let Q=pZ (0), for some prime number p. Then Q is a classical quasi primary element of L(M) but it is not a primary element of L(M) see[1]. In the next result we obtain characterizations of primary elements, classical primary elements and classical quasi primary elements of a multiplicative lattice. Theorem 1. Consider the following statements for a proper element q of L, (i) q is a primary element (ii) q is a classical primary element (iii) (q : c) is a primary element for each element c of L such that c q (iv) q is a classical quasi primary element

4 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), (v) (q : c) is a prime element for each element c of L such that c q (vi) q is a quasi primary element that is q= (q : 1) is a prime element (vii) q is a power of prime element. Then(i) (ii),(ii) (iii),(iii) (iv),(iv) (v),(v) (vi),(v) (iv),(vii) (vi). Proof.(i) (ii) Suppose, q is a primary element. Let a, b L, r Lsuch that abr q and br q. Since, q is a primary element, abr q,br q implies a k q for some positive integer k. Hence, a k r q. Thus, q is a classical primary element of L. (ii) (iii) Suppose, q is a classical primary element of L and let ab(q : c), a, b L. Then(ab)c q. Hence, ac q or b k c q, that is a(q : c) or b k (q : c) for some positive integer k. Therefore,(q : c) is a primary element for each c q. (iii) (i) Suppose,(q : c) is a primary element of L, for each c q. Take c= 1. Then,(q : 1) is primary. Let abq. So,(ab)(q : 1) and a(q : 1) or b k (q : 1). Hence, aq or b k q and q is a primary element. That is(i) (ii) (iii) (i). (iii) (v) Suppose,(q : c) is primary, for any c q. But(q : c) is a primary implies (q : c) is prime. (iv) (v) Suppose, q is a classical quasi primary element and let ab (q : c) where c q. So, (ab) k (q : c) for some positive integer k. That is a k b k cq. As q is classical primary, (a k ) t c q or(b k ) t c q. That is a m c q or b m c q where m=kt Z +. This shows that, a (q : c) or b (q : c). Therefore, (q : c) is prime. (v) (iv) Suppose, (q : c) is prime for any c q. Let abr q where a, b, r L. We have, ab(q : r) (q : r). Let r q. Then (q : r) is prime implies a (q : r) or b (q : r). This implies, a n r q or b n r q for some positive integer n. Thus, q is a classical quasi primary element. (v) (vi) Suppose, (q : c) is prime, c q. Let ab q. Then(ab) k q for some positive integer k. Take c= 1. Then, (q : 1) is prime and(ab) k (q : 1), that is ab (q : 1) which is prime. Hence, a (q : 1) or b (q : 1). Therefore, a q or b q and q is prime. (vii) (vi) Suppose, q is a power of a prime element and q= p k where p is prime and k is positive integer. We prove that, q is prime. Let ab q. Then(ab) m q= p k p, for some positive integer m. Then a m p or b m p. So, a p or b p and hence, a k p k or b k p k = q. This shows that a q or b q and q is a prime element. We now prove the characterizations of a classical primary element and a classical quasi primary element of a lattice module M. These results establish the relation between a classical

5 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), primary element of a lattice module and a primary element of a multiplicative lattice and also the relation between a classical quasi primary element of a lattice module and a quasi primary element of a multiplicative lattice. Theorem 2. Let M be a lattice module and Q be a proper element of M. Then (i) Q is a classical primary element if and only if for every element N of M such that N Q,(Q : N) is a primary element of L. (ii) Q is a classical quasi primary if and only if for every element N M such that N Q, (Q : N) is quasi primary element of L. Proof. (i) Suppose, Q is a classical primary element of M and N Q. Let ab(q : N). Since, Q is a classical primary element of M,aN Q or b k N Q for some positive integer k. That is a(q : N) or b k (Q : N). Hence,(Q : N) is a primary element of L. Conversely, suppose(q : N) is a primary element of L for any N M such that N Q. Let abn Q. Then, ab(q : N), which is primary. So, a(q : N) or b k (Q : N). That is an Q or b k N Q. Suppose, N Q. Then for any a L, an N Q implies(q : N)=1and hence ab(q : N)=1. So, a(q : N), b(q : N) which shows that an Q and bn Q, when abn Q. Hence, Q is a classical primary element of M. (ii) Let Q be a classical quasi primary element and let ab (Q : N) where N Q. Then, (ab) k (Q : N) for some positive integer k. Since, Q is a classical quasi primary element of M, a k b k N Q implies(a k ) l N Q or(b k ) l N Q. That is a n N Q or b n N Q for some n Z +. Hence, a (Q : N) or b (Q : N) and (Q : N) is prime. Thus,(Q : N) is a quasi primary element of L. Suppose,(Q : N) is a quasi primary element. Let abn Q where a, b L and for each element N Q. Then ab(q : N) (Q : N)). Since,(Q : N) is quasi primary, (Q : N) is prime. So that a k (Q : N) or b k (Q : N). That is, a k N Q or b k N Q. Hence, Q is a classical quasi primary element. If N Q,(Q : N)=1. In this case abn Q implies ab(q : N)=1and obviously a(q : N), b(q : N). Consequently, an Q, bn Q and Q is a classical quasi primary element. The following theorem is obvious. Theorem 3. Let M be a L-module and Q be a proper element of M. (i) The following statements are equivalent, (a) Q is a classical primary element. (b) For every a, b L and N M, abn Q implies that either an Q or b k N Q for some k Z + (c) For every N M where N Q,(Q : N) is a primary element of L. (ii) The following statements are equivalent. (a) Q is a classical quasi primary element. (b) For every a, b L and N M, abn Q implies either a k N Q or b k N Q for some k Z +

6 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), (c) For every N M, where N Q,(Q : N) is a quasi primary element of L. Theorem 4. Let M be a lattice module and Q be a classical primary (or classical quasi primary) element of M. Then{ (Q : N) N Q} is a chain of prime element of L. Proof. For each M 1, M 2 Q, we show that (Q : M 1 ) (Q : M 2 ) (Q :(M 1 M 2 )). Let x (Q : M 1 ) (Q : M 2 ). Then x (Q : M 1 ) and x (Q : M 2 ). So x n 1 M 1 Q, x n 2 M 2 Q for some integers n 1, n 2 and let n= max{n 1, n 2 }. Hence x n (M 1 M 2 )Q. That is x (Q :(M 1 M 2 )) and we have (Q : M1 ) (Q : M 2 ) (Q :(M 1 M 2 )). Since Q is classical primary and M 1 M 2 Q, (Q :(M 1 M 2 )) is prime by Theorem 1, we conclude that, (Q : M 1 ) (Q :(M 1 M 2 )) or (Q : M 2 ) (Q :(M 1 M 2 )). Suppose (Q : M1 ) (Q :(M 1 M 2 )). Then {x L x k 1 M 1 Q, k 1 Z + } {x L x k 2 M 1 x k 2 M 2 Q} = {x L x k 2 M 1 and x k 2 M 2 Q}. Hence x k 1 M 1 Q implies x k 2 M 2 Q for some integers k 1, k 2. Therefore, {x L x k 1 M 1 Q} {x L x k 2 M 2 Q}. Consequently, (Q : M 1 ) (Q : M 2 ). Similarly, (Q : M 2 ) (Q :(M 1 M 2 )) implies (Q : M2 ) (Q : M 1 ). Thus (Q : M 1 ) (Q : M 2 )or (Q : M 2 ) (Q : M 1 ). Hence { (Q : N) N Q} is a chain of prime elements of L. We now obtain the characterizations of a classical primary element, a primary element of a lattice module and a primary element of a multiplicative lattice. Theorem 5. Let M be a multiplication L-module and Q be a proper element of M. The following statements are equivalent, (i) Q is a classical primary element of M. (ii) Q is a primary element (iii) (Q : I M ) is a primary element of L (iv) Q=qI M where q is primary element of L, is maximal with respect to this property i.e. ai M = Q implies aq, a L. Proof.(i) (ii) Assume that Q is a classical primary element of M. Let, an Q, a L, N M and N Q. Since, M is a multiplication module, N= bi M for some b L. Hence, abi M Q and bi M Q, i.e. ab(q : I M ) and b (Q : I M ). By Theorem 3,(Q : I M ) is a primary element of L. Hence, a k I M Q for some k Z +. Thus, Q is primary element of M.

7 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), (ii) (iii) Assume that, Q is a primary element of M. Let ab(q : I M ). Then, abi M Q. Since, Q is a primary element of M, either bi M or a k I M Q. So, b(q : I M ) or a k (Q : I M ) and (Q : I M ) is a primary element of L. (iii) (iv) Assume that, q=(q : I M ) is a primary element of L. Since, M is a multiplication module, Q=aI M for somea L. We have, qi M Q and a(q : I M )=q. So Q=aI M qi M. Thus, Q=qI M,q is primary and bi M = Q bq. (iv) (i) Suppose, Q=qI M where q is primary element which is maximal w.r.t. this property. Let, abn Q where a, b L and N such that bn Q. Since, M is a multiplication L-module N= ci M for somec L. Thus abci M Q and hence abc(q : I M )q. Since, bci M Q, we have, bc (Q : I M )q. As, q is primary and abc q withbc q we have, a k q for some k Z +. This implies that, a k N qi M = Q and so Q is a classical primary element of M. In the next result we have the important relation between a classical quasi primary element of a lattice module and a classical quasi primary element of a multiplicative lattice. Theorem 6. Let M be a multiplication lattice module and Q be a proper element of M. The following statements are equivalent, (i) Q is a classical quasi primary element. (ii) q=(q : I M ) is a classical quasi primary element of L. (iii) Q=qI M where q is a classical quasi primary element which is maximal w.r.t this property. (i.e. ai M = Q aq) Proof.(i) (ii) Suppose, Q is a classical quasi primary element of M. Let, abr q, a, b, r L. This gives, abri M Q. By classical quasi primality of Q, we have, either(ab) k I M Q or r k I M Q for some k Z +. i.e. (ab) k (Q : I M ) or r k (Q : I M ). As, Q is a classical quasi primary element and I M Q,(Q : I M ) is a quasi primary element of L i.e. (Q : IM ) is a prime element of L, by Theorem 5. Hence abri M Q implies ab (Q : I M ) or r (Q : I M ). Thus again, a (Q : I M ) or b Q : I M. So a k (Q : I M ) or b k (Q : I M ). Consequently, a k r(q : I M ) or b k r(q : I M ) i.e. a k r q or b k r q. Hence, q is classical quasi primary. (ii) (iii) Suppose, q=(q : I M ) is a classical quasi primary element of L. Since, M is a multiplication module, Q=aI M for some a L. We have, qi M Q and a(q : I M )=q. Thus, ai M = Q gives a q. So, q is maximal w.r.t. Q=qI M,(q L) (iii) (i) Suppose, Q=qI M where q is classical quasi primary element which is maximal w.r.t. this property. So, ai M = Q implies aq. We show that, Q is a classical quasi primary element of M. Let abn Q where a, b L, N M. Suppose, b k N Q for any integer k. Since M is a

8 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), multiplication L-module. N= ci M for some c L. Thus, abci M Q i.e. abc(q : I M )q where q is a classical quasi primary element of L. Now, abc q and b k ci M Q implies b k c (Q : I M )=q for any k. Therefore we have a k c q for some integer k. So, a k ci M qi M, i.e. a k N qi M = Q. Consequently, Q is a classical quasi primary element of M. 3. Classical Quasi Primary Decomposition of Elements The study of decomposition into classical primary submodules was introduced in[3]. Further investigation of decomposition of submodules into classical primary submodules is carried out by Behboodi et al.[1]. We carry out this study for lattice modules. The next result shows when the element having a primary decomposition is classical quasi primary element. Theorem 7. Let M be an L-module and let Q= Q 1 Q 2 Q n be a primary decomposition of Q with p i = (Q i : I M ). If p 1 p 2 p n then Q is a classical p 1 -quasi primary element. Proof. Assume that, abn Q, where a, b L, N Q. Then, N Q i, for some i(1 in). Suppose, t(1 t n) is the smallest number such that N Q t. Thus, N Q 1 Q 2 Q t 1. We have abn Q t and Q t is p t primary. Hence,(ab) k 1 I M Q t for some k 1 Z + implies ab (Q t : I M )= p t. Thus, ap t or bp t. Now p t p t+1 p n. Suppose, ap t = (Q t : I M ). Consequently a k I M Q t, for some integer k. If bp t = (Q t : I M ) we have b k I M Q t. Thus a k I M Q t or b k I M Q t for some k Z + where t is the smallest positive integer such that N Q t. We have N Q t+1,..., N Q n, a k I M Q t,q t+1,...,q n or b k I M Q t,q t+1,...,q n. Hence a k I M Q t Q t+1 Q n or b k I M Q t Q t+1 Q n for some k Z +. But, N Q 1 Q 2 Q t 1. It follows that, a k N Q 1 Q 2 Q n = Q or b k N Q 1 Q 2 Q n. If N Q, abn Q implies an Q, bn Q. Also, (Q : IM )= (Q 1 Q 2 Q n ) : I M = p 1. Therefore, Q is a classical p 1 - quasi primary element. Remark 1. The following example shows that the above theorem is not necessarily true if Q 1,Q 2,...,Q n are only assured to be classical (quasi) primary elements. Example 5. Let R= Z, M= Z 2 Z 3 Z Then M is a z-module. Let L(R) denote the set of all ideals of R and L(M) denote the set of all submodules of M. Then L(M) is a module over L(R) where L(R) is a multiplicative lattice. Let Q 1 = Z 2 (0) (0). Q 2 =(0) Z 3 (0). Then Q 1, Q 2, are elements of L(M). It can be shown that Q 1 and Q 2 are classical (quasi) primary elements of L(M). Also,(0)=Q 1 Q 2 and (Q 1 : M)= (Q 2 : M)=(0). Obviously, 2 3(Z 2 Z 3 (0))=(0). Also, for each k 1,2 k (Z 2 Z 3 (0)) (0) and 3 k (Z 2 Z 3 (0)) (0). Thus,(0) M is not a classical (quasi) primary element. Remark 2. We shall show that the converse of Theorem 7 is also true when decomposition Q 1 Q 2 Q n is a reduced primary decomposition. Theorem 8. Let Q be a p-primary element of a lattice module M and N be an element of M. If N Q then(q : N) is a p-primary element.

9 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), Proof. First we show that(q : N) is a primary element. Let a, b L, ab(q : N) and suppose a (Q : N). Also as ab(q : N), abn Q with an Q and Q is a primary element implies that b n (Q : I M ) for some integer n. But b n I M Q implies b n N Q and we have b (Q : N). Therefore,(Q : N) is a primary element of L. Now since N Q, there exists A I M and A N such that A Q. Let a (Q : N). Then a n N Q and a n A Q. But A Q and Q is primary implies that(a n ) k = a m (Q : I M ) for some integer m. That is a (Q : I M )= p and (Q : N) p. Conversely, let a (Q : I M )= p. Hence, a n I M Q for some integern. So a n N Q for some integer n. Thus a n (Q : N) and a (Q : N). This shows that p (Q : N) and we have (Q : N)= p. Therefore,(Q : N) is a p-primary element. We now establish the characterization of an associated prime of an element of a lattice module having a primary decomposition. Theorem 9. Let N I M be an element of a lattice module M and assume that N has a primary decomposition. Let N= Q 1 Q 2 Q k be a reduced primary decomposition of N and p be prime element of L. Then following statements are equivalent, (i) p= Q i for some i (ii) (N : X) is a p-primary element of L for some X N. Proof.(i) (ii) Let N = Q 1 Q 2 Q k be a reduced primary decomposition of N. First suppose that, p= Q i for some i. Without loss of generality we can assume that p= (Q 1 : I M ) where p i = (Q i : I M ) i= 1, 2,..., k. We prove that,(n : X) is a p-primary element of L for some X N. Since the decomposition is reduced Q i Q 1 Q 2 Q i 1 Q i+1 Q k for i= 1, 2,..., k. In particular, Q 1 Q 2 Q 3 Q k. So there exists X Q 2 Q 3 Q k such that X Q 1 and hence X N= Q 1 Q 2 Q k. Also (N : X)=(Q 1 Q 2 Q k ) : X=(Q 1 : X) (Q 2 : X) (Q k : X). For i= 2, 3,..., k we show that(q i : X)=1. Since X Q 2 Q 3 Q k, we have X Q i for all i= 2,..., k. Then ax Q i for all a L and for all i= 2,3,..., k. That is a(q i : X) for all i= 2, 3,..., k. So 1(Q i : X). But(Q i : X)1implies(Q i : X)=1for i= 2,3,..., k. Hence,(N : X)=(Q 1 : X) 1 1=(Q 1 : X). So by the above result,(q 1 : X) is p-primary element implies(n : X) is a p-primary element of L where X N. (ii) (i) Assume that(n : X) is a p-primary element of L for some X N, X M. We prove that Qi = p for some i. We have, p= (N : X)= [(Q 1 Q 2 Q k ) : X]= (Q 1 : X) (Q 2 : X) (Q k : X). We claim that for each i, (Q i : X)= p i or 1 and equal to p i for at least one i. We have, X N= Q 1 Q 2 Q k implies X Q i for at least one i(1ik). Suppose, X Q r (1 r k) and X Q 1 Q 2 Q r 1 Q r+1...q k that is X Q i, where(i r).

10 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), Let a(q i : X), a L. Since, X Q i. We have, ax Q i for all i r and a L. Hence, a (Q i : X) for all a L. In particular, 1 (Q i : X) for all i r. But, (Q i : X)1 for all i r. Therefore, (Q i : X)=1 for all i r. For i=r, X Q r. Let a (Q r : X). Hence, a n X Q r, for some positive integer n, where X Q r. As Q r is primary, a n (Q r : I M )= p r. Thus, ap r, since p r is prime and we have, (Q r : X) p r. On the other hand, let a p r = Q r = (Q r : I M ). Hence, a n (Q r : I M ) for some positive integer n. That is a n I M Q r and therefore, a n X Q r, for some positive integer n. Consequently, a n (Q r : X) and hence a (Q r : X). This gives p r (Q r : X). Hence, (Q r : X)= p r where X Q r. We have shown that for each i, (Q i : X)= p i or 1 and is equal to p i for at least one i, since X N. Then, p= (N : X)= (Q 1 : X) (Q k : X) is the meet of some of the prime elements p 1, p 2,..., p l (1 l k). That is p= (N : X)= p 1 p 2 p l. We show that p= p i for some i. We have, p p i i= 1,2,..., l. If for each i p p i then p i p for all i= 1,2,..., l. This implies that there exist x i p i such that x i p for all i= 1,2,..., l. Then, x 1 x 2... x l p 1 p 2 p l = p. This shows that x i p for at least one i(1 ik) a contradiction. Hence, p= p i for at least one i. We have the characterization of a classical quasi primary element of a lattice module in terms of a chain of its associated primes. Theorem 10. Let M be a L-module and Q be a proper element of M. Let Q= Q 1 Q 2 Q n with p i = (Q i : I M ) be a reduced primary decomposition of Q. Then Q is a classical quasi primary element if and only if{p 1, p 2,..., p n } is a chain of prime elements of L. In that case, radical of (Q : I M ) is the smallest of the primes p 1, p 2,..., p n. Proof. Since, Q= Q 1 Q 2 Q n is a reduced primary decomposition of Q by Theorem 7 for each i(1in), p i = (Q : A i ) for some A i Q where p i = (Q i : I M ). Assume that, Q is a classical quasi primary element. We show that{p 1, p 2,..., p n } is the chain of prime elements of L. Assume that, it is not a chain. Then p i p j and p j p i for some i j. Select ap i such that a p j and bp j such that b p i. i.e. a (Q i : I M ), a (Q j : I M ) and b (Q j : I M ), b (Q i : I M ). Since p i = (Q : A i ), A i Q for each i= 1,2,..., n, we have, a (Q i : I M )= (Q : A i ) and a (Q : A j ). Similarly, b (Q j : I M )= (Q : A j ) and b (Q : A i ). This shows that a k ia i Q for some integer k i and a k A j Q for any k. Similarly, b k ja j Q for some integer k j and b k A i Q for any k. Thus a k ia i b k ja j Q and a k A j b k A i Q for any k. Let k=max{k i, k j }. Then a k b k A i b k a k A j = a k b k (A j A i ) Q. Since, Q is a classical quasi primary element of M, it follows that(a k ) l (A i A j )Q or (b k ) l (A i A j )Qi.e. a s (A i A j )Qor b s (A i A j )Qfor some integer s. Therefore, a s (Q : A j ) or b s (Q : A i ) for some integer s. Hence, ap j or bp i, which is a contradiction. Thus,{p 1, p 2,..., p n } is a chain of prime elements of L. Converse follows by Theorem 7. Remark 3. We note that Theorem 8 is not necessarily true if the primary decomposition Q= Q 1 Q 2 Q n is not minimal. Example 6. Let R= Z, M= Z Z. Let L(R) denote the set of all ideals of R and L(M) denote the set of all sub modules of a module M over R= Z. Then L(M) is an L-module over L(R). Let

11 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), Q 1 = 2Z Z, Q 2 = Z 3Z, Q 3 = Z (0), Q 4 =(0) Z. Then, Q 1, Q 2, Q 3, Q 4 are primary sub modules of M and hence, these are the elements of a lattice module L(M) with (Q i : M)=2Z, (Q2 : M)=3Z and (Q i : M)= (Q 4 : M)=(0). Also,(0)=Q 1 Q 2 Q 3 Q 4 and (0) is a classical quasi primary sub module of M and hence a classical quasi primary element of L(M). But {(0), 2Z, 3Z} is not a chain of prime ideals of R i.e.(0),2z,3z is not a chain of prime elements of L(R). Definition 7. Let N be a proper element of a lattice module M over L. A classical primary (respectively classical quasi primary) decomposition of N is an expression N= n i=1 Q i where each Q i is classical primary (respectively classical quasi primary) element of M.The decomposition is called reduced if it satisfies the following two conditions, (i) No Q i1 Q i2 Q it is classical primary (respectively classical quasi primary) element where(i 1, i 2,..., i t ) {1, 2,..., n} for t 2 with i 1 < i 2 < < i t (ii) for each j, Q j i j Q i Corresponding to the above definition by Theorem 2 we have a list of prime elements (Q1 : I M ),... (Q n : I M ). Among reduced classical primary (resp classical quasi primary) decomposition, any one that has the least number of distinct primes will be called minimal. It is clear that, every primary decomposition of an element N of M is called classical primary but the converse is not always true. The next result is useful in proving the uniqueness of associated primes of an element of a lattice module with a primary decomposition. Theorem 11. Let N= Q 1 Q 2 Q k be a reduced primary decomposition of a lattice module M over L. Then every minimal prime divisor of N is a prime divisor of N. Proof. Let p i = (Q i : I M ). Then p 1, p 2,..., p k are the prime elements which are called prime divisors of N or associated primes of N. We have (N : IM )= (Q 1 Q 2 Q k ) : I M = (Q 1 : I M ) (Q 2 : I M ) (Q k : I M ) =p 1 p 2 p k. If a p 1 p 2 p 3... p k then a p i for each i= 1,2,..., n. Hence a (Q i : I M ) for each i. That is a (Q 1 : I M ) (Q 2 : I M )... (Q k : I M )= (N : I M ). So a n (N : I M ) for some positive integer n. If p is any prime element of L containing(n : I M ) then a n p and hence a p. Then p 1 p 2... p k p whenever(n : I M ) p. This shows that p i p for some i. If p is a minimal prime element containing(n : I M ) then p i p for some i. Also p i contains(n : I M ), but p is minimal such element. Hence p i = p. Thus every minimal prime divisors of N is a prime divisor of N and is minimal in the set of prime divisors of N. The next result we prove the important property of uniqueness of associated primes of an element having classical quasi primary decomposition.

12 C. Manjarekar, U. Kandale/Eur. J. Pure Appl. Math, 8 (2015), Theorem 12. Let L be a Noetherian lattice and M be a module over L. Let N be a proper element of M and N= Q 1 Q 2 Q n with (Q i : I M )= p i for i= 1,2,..., n be a reduced classical quasi primary decomposition of N. Then,{p i i= 1,2,..., n}= min(n : I M )} and the set {p i i= 1, 2,..., n} is uniquely determined. Proof. First we show that, min(n : I M ) {p i i= 1,2,3,..., n}. Let p be a minimal prime of(n : I M ). Then p is a minimal member of associated primes of N. But p=p i for some i if and only if there exists A N in M such that(n : A) p-primary. Thus, p= (N : A)= (Q : I M ) for some A N. Renumber the Q i such that A Q i for 1i j and AQ i for j+ 1in. Since p i = (Q i : I M ), p i k i I M Q i for some integer k i, (1 in). Therefore,( j k p i i )A n Q i = N and so j k p i i (N : A) p. Since p is prime, i=1 i=1 i=1 p t p for some t j. As,(N : I M ) (N : I M ) (Q t : I M )= p t and p is a minimal prime of(n : I M ), we conclude that p= p t. Now it is sufficient to show that each p i (1 in) is a minimal prime of(n : I M ). Without loss of generality, we may take i= 1. Clearly (N : I M ) (N : I M )= (Q 1 Q 2 Q n : M)= n i=1 (Qi : I M ) p 1. On the contrary, suppose that p 1 is not a minimal prime of(n : I M ). Thus an i {1,2,..., n} such that p i is minimal prime of(n : I M ) with p i < p 1 (since min(n : I M ) {p i i= 1, 2, 3,..., n}). Again without loss of generality, we may take i= 2. Thus(N : I M ) p 2 < p 1. By[5], each Q i has a minimal primary decomposition. Suppose that, Q 1 = Q 11 Q 1s with (Q 1j : I M )= p 1j (1 j s) and Q 2 = Q 21 Q 2t with (Q2j : I M )= p 2j (1 jt) are a reduced primary decompositions of Q 1 and Q 2 respectively. By Theorem 8{p 1j 1 j s} and{p 2j 1 jt} are chains of prime elements. Without loss of generality, we can assume that, p 11 p 12 p 1s and p 21 p 22 p 2t. We thus get p 1 = p 11 and p 2 = p 21, since and p 1 = (Q 1 : I M )= (Q 11 Q 1s ) : I M = s i=1 (Q1i : I M )= p 11 p 2 = (Q 2 : I M )= (Q 21 Q 2t ) : I M = t i=1 (Q2i : I M )= p 21. It follows that, p 21 p 11 p 12 p 1s and so by Theorem 8, Q 1 = Q 21 Q 11 Q 1s is a classical quasi primary element of M with (Q 1 : I M )= p 21 = p 2. On the other hand, N=Q 1 Q n =(Q 11 Q 1s ) (Q 21 Q 2t ) Q 3 Q n =(Q 21 Q 11 Q 1s ) (Q 21 Q 2t ) (Q 3 Q n ). Thus, N= Q 1 Q 2 Q n is a classical quasi primary decomposition of N with (Q 1 : I M)= (Q 2 : I M )= p 2 and (Q i : I M )= p i for i= 3,... n. We note that if another Q i (3in) such that (Q i : I M )= p i = p 1 then by similar arguments we can replace it

13 REFERENCES 184 by Q i such that (Q i : I M)= (Q 2 : I M )= p 2. Now, by using this decomposition we can obtain a reduced classical quasi primary decomposition, N = Q 1 Q 2 Q such that k p 1 / { (Q : I i M ) i= 1,2, 3, 4,..., k} {p i i= 2,..., n}, contrary with the irrenduntness of the decomposition N= Q 1, Q 2..., Q n with { (Q i : I M ) i= 1, 2,3,..., n}={p i i= 1,2..., n}. Thus,{p i i= 1, 2,..., n}= min(n : I M ). ACKNOWLEDGEMENTS The authors thank the readers of European Journal of Pure and Applied Mathematics, for making our journal successful. We dedicate this research article to Prof Dr N K Thakare on his 76 birthday. References [1] M. Behboodi, R. Jahani-nezhad, and M. H. Naderi. Classical Quasi Primary Submodules,Bulletin of the Iranian Mathematical Society, 37(4), [2] M. Behboodi and M. Baziar. Classical primary submodules and decomposition theory of modules, Journal of Algebra Application, 8(3), [3] R. P. Dilworth. Abstract Commutative Ideal Theory, Pacific J. Math., 12, [4] L. Fuchs. On quasi primary ideals, Acta Scientiarum Mathematicarum, 11, [5] R. Y. Sharp. Steps in commutative algebra, London Mathematatical Society, Cambridge University Press, Cambridge, 1990.