/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 8, No. 2, 2015, 172-184

ISSN 1307-5543 – www.ejpam.com

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 ele-

ments 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 de-

composition and minimal decomposition into classical quasi primary elements.

2010 Mathematics Subject Classifications: 13A99

Key Words and Phrases: Primary element, prime element,classical quasi primary element,lattice mod-

ules.

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 ab ¶ p implies a ¶ p or b ¶ p for a, b ∈ L

and is called primary element if ab ¶ p implies a ¶ p or bn ¶ p for some positive integer

n. For a ∈ L,
p

a = ∨{x ∈ L | xn ¶ 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

aα)A= ∨
α

aαA ∀aα ∈ L, A∈ M for some integer

(ii) a(∨
α

Aα) = ∨
α

aAα ∀a ∈ L, Aα ∈ M

(iii) (ab)A= a(bA) ∀a, b ∈ L, A∈ M

(iv) IA= A ∀A∈ M

(v) OA= OM ∀A∈ M where OM = gl b(M).

∗Corresponding author.

Email addresses: csmanjrekar@yahoo.co.in (C. Manjarekar), ujwalabiraje@gmail.com (U. Kandale)

http://www.ejpam.com 172 c© 2015 EJPAM All rights reserved.



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 173

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 OM and IM . 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 cb ¶ a,

• 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 6= IM of a lattice module M is called a prime element if whenever aA¶ N where

a ∈ L,A ∈ M implies either a ¶ (N : IM ) or A ¶ N . An element N 6= IM of a lattice module

M is called a primary element if whenever aA ¶ N where a ∈ L,A ∈ M implies either A ¶ N

or an ¶ (N : IM ) 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 = aIM .An

element N 6= IM of a lattice module M is said to have primary decomposition if there exist

primary elements Q1,Q2, . . . ,Qk such that N = Q1 ∧Q2 ∧ · · · ∧ Qk. If some Q i contains the

meet of remaining ones then this Q i can be dropped from the primary decomposition. Simi-

larly 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 de-

composition 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 ∈ L implies that either ar ¶ q or bkr ¶ 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 ∈ L implies that either akr ¶ q or bkr ¶ 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 ¶
p

q where a, b ∈ L implies that

either ak ¶ q or bk ¶ q for some integer k.



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 174

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 bkN ¶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 akN ¶Q or bkN ¶Q for some integer k.

Definition 6. A proper element Q of M is called a quasi primary element if
p

(Q : IM ) 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∈∧ Rλ 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
pk + 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,
p

(N : Q) = (0). If

N = Z + Z .(1
5), the submodule of M generated by {1, 1

5}, then 2.3〈 1
2.3〉 ⊆ N, but for each k ≥ 1,

2k〈 1
2.3〉 6⊆ N and 3k〈 1

2.3〉 6⊆ 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 ele-

ments 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



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 175

(v)
p

(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
p

q =
p

(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 ∈ L such that abr ¶ q and br 6¶ q. Since,

q is a primary element, abr ¶ q,br 6¶ q implies ak ¶ q for some positive integer k. Hence,

akr ¶ 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 bkc ¶ q, that is a ¶ (q : c) or bk ¶ (q : c) for some positive integer k.

Therefore, (q : c) is a primary element for each c 6¶ q.

(iii)⇒ (i)
Suppose, (q : c) is a primary element of L, for each c 6¶ q. Take c = 1. Then, (q : 1) is primary.

Let ab ¶ q. So, (ab) ¶ (q : 1) and a ¶ (q : 1) or bk ¶ (q : 1). Hence, a ¶ q or bk ¶ q and q is

a primary element.

That is (i)⇒ (ii)⇒ (iii)⇒ (i).
(iii)⇒ (v)

Suppose, (q : c) is primary, for any c 6¶ q. But (q : c) is a primary implies
p

(q : c) is prime.

(iv)⇒ (v)
Suppose, q is a classical quasi primary element and let ab ¶

p

(q : c) where c 6¶ q. So,

(ab)k ¶ (q : c) for some positive integer k. That is ak bkc ¶ q. As q is classical primary,

(ak)t c ¶ q or (bk)t c ¶ q. That is amc ¶ q or bmc ¶ q where m = kt ∈ Z+. This shows that,

a ¶
p

(q : c) or b ¶
p

(q : c). Therefore,
p

(q : c) is prime.

(v)⇒ (iv)
Suppose,
p

(q : c) is prime for any c 6¶ q. Let abr ¶ q where a, b, r ∈ L. We have,

ab ¶ (q : r)¶
p

(q : r). Let r 6¶ q. Then
p

(q : r) is prime implies a ¶
p

(q : r) or

b ¶
p

(q : r). This implies, anr ¶ q or bnr ¶ q for some positive integer n. Thus, q is a

classical quasi primary element.

(v)⇒ (vi)

Suppose,
p

(q : c) is prime, c 6¶ q. Let ab ¶
p

q. Then (ab)k ¶ q for some positive integer k.

Take c = 1. Then,
p

(q : 1) is prime and (ab)k ¶ (q : 1), that is ab ¶
p

(q : 1) which is prime.

Hence, a ¶
p

(q : 1) or b ¶
p

(q : 1). Therefore, a ¶
p

q or b ¶
p

q and
p

q is prime.

(vii)⇒ (vi)

Suppose, q is a power of a prime element and q = pk where p is prime and k is positive integer.

We prove that,
p

q is prime. Let ab ¶
p

q. Then (ab)m ¶ q = pk ¶ p, for some positive integer

m. Then am ¶ p or bm ¶ p. So, a ¶ p or b ¶ p and hence, ak ¶ pk or bk ¶ pk = q. This shows

that a ¶
p

q or b ¶
p

q and
p

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



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 176

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 6¶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 6¶ Q,

(Q : N) is quasi primary element of L.

Proof. (i) Suppose, Q is a classical primary element of M and N 6¶ Q. Let ab ¶ (Q : N).

Since, Q is a classical primary element of M,aN ¶ Q or bkN ¶ Q for some positive integer k.

That is a ¶ (Q : N) or bk ¶ (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 6¶ Q. Let abN ¶ Q.

Then, ab ¶ (Q : N), which is primary. So, a ¶ (Q : N) or bk ¶ (Q : N). That is aN ¶ Q or

bkN ¶ Q. Suppose, N ¶ Q. Then for any a ∈ L, aN ¶ N ¶ Q implies (Q : N) = 1 and 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 ¶
p

(Q : N) where N 6¶ Q. Then,

(ab)k ¶ (Q : N) for some positive integer k. Since, Q is a classical quasi primary element of

M, ak bkN ¶ Q implies (ak)l N ¶ Q or (bk)l N ¶ Q. That is anN ¶ Q or bnN ¶ Q for some

n ∈ Z+. Hence, a ¶
p

(Q : N) or b ¶
p

(Q : N) and
p

(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 6¶ Q. Then ab ¶ (Q : N) ¶
p

(Q : N)). Since, (Q : N) is

quasi primary,
p

(Q : N) is prime. So that ak ¶ (Q : N) or bk ¶ (Q : N). That is, akN ¶ Q or

bkN ¶ Q. Hence, Q is a classical quasi primary element. If N ¶ Q, (Q : N) = 1. In this case

abN ¶ Q implies ab ¶ (Q : N) = 1 and 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 bkN ¶ Q for

some k ∈ Z+

(c) For every N ∈ M where N 6¶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 akN ¶ Q or bkN ¶ Q for

some k ∈ Z+



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 177

(c) For every N ∈ M, where N 6¶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 {p(Q : N) | N 6¶Q} is a chain of prime element of L.

Proof. For each M1, M2 6¶ Q, we show that
p

(Q : M1) ∧
p

(Q : M2) ¶
p

(Q : (M1 ∨M2)).

Let x ¶
p

(Q : M1)∧
p

(Q : M2). Then x ¶
p

(Q : M1) and x ¶
p

(Q : M2). So

xn1 M1 ¶Q, xn2 M2 ¶Q for some integers n1, n2 and let n= max{n1, n2}. Hence

xn(M1 ∨M2)¶Q. That is x ¶
p

(Q : (M1 ∨M2)) and we have

Æ

(Q : M1)∧
Æ

(Q : M2)¶
Æ

(Q : (M1 ∨M2)).

Since Q is classical primary and M1 ∨ M2 6¶ Q,
p

(Q : (M1 ∨M2)) is prime by Theorem 1, we

conclude that,
p

(Q : M1) ¶
p

(Q : (M1 ∨M2)) or
p

(Q : M2) ¶
p

(Q : (M1 ∨M2)). Suppose
p

(Q : M1)¶
p

(Q : (M1 ∨M2)). Then

∨{x ∈ L | xk1 M1 ¶Q, k1 ∈ Z+}¶ ∨{x ∈ L | xk2 M1 ∨ xk2 M2 ¶Q}
= ∨{x ∈ L | xk2 M1 ¶ and xk2 M2 ¶Q}.

Hence xk1 M1 ¶Q implies xk2 M2 ¶Q for some integers k1, k2. Therefore,

∨{x ∈ L | xk1 M1 ¶Q}¶ ∨{x ∈ L | xk2 M2 ¶Q}.

Consequently,
p

(Q : M1) ¶
p

(Q : M2). Similarly,
p

(Q : M2) ¶
p

(Q : (M1 ∨M2)) implies
p

(Q : M2) ¶
p

(Q : M1). Thus
p

(Q : M1) ¶
p

(Q : M2)or
p

(Q : M2) ¶
p

(Q : M1). Hence

{p(Q : N) | N 6¶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 : IM ) is a primary element of L

(iv) Q = qIM where q is primary element of L, is maximal with respect to this property i.e.

aIM =Q implies a ¶ q, a ∈ L.

Proof. (i)⇒ (ii)
Assume that Q is a classical primary element of M. Let, aN ¶ Q, a ∈ L, N ∈ M and N 6¶ Q.

Since, M is a multiplication module, N = bIM for some b ∈ L. Hence, abIM ¶Q and bIM 6¶Q,

i.e. ab ¶ (Q : IM ) and b 6¶ (Q : IM ). By Theorem 3, (Q : IM ) is a primary element of L. Hence,

ak IM ¶Q for some k ∈ Z+. Thus, Q is primary element of M.



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 178

(ii)⇒ (iii)
Assume that, Q is a primary element of M. Let ab ¶ (Q : IM ). Then, abIM ¶ Q. Since, Q is

a primary element of M, either bIM ¶ or ak IM ¶ Q. So, b ¶ (Q : IM ) or ak ¶ (Q : IM ) and

(Q : IM ) is a primary element of L.

(iii)⇒ (iv)
Assume that, q = (Q : IM ) is a primary element of L. Since, M is a multiplication module,

Q = aIM for somea ∈ L. We have, qIM ¶ Q and a ¶ (Q : IM ) = q. So Q = aIM ¶ qIM . Thus,

Q = qIM ,q is primary and bIM =Q⇒ b ¶ q.

(iv)⇒ (i)
Suppose, Q = qIM 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 6¶ Q. Since, M is a multiplication L-module

N = cIM for somec ∈ L. Thus abcIM ¶ Q and hence abc ¶ (Q : IM ) ¶ q. Since, bcIM 6¶ Q,

we have, bc 6¶ (Q : IM ) ¶ q. As, q is primary and abc ¶ q withbc 6¶ q we have, ak ¶ q for

some k ∈ Z+. This implies that, akN ¶ qIM = 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 : IM ) is a classical quasi primary element of L.

(iii) Q = qIM where q is a classical quasi primary element which is maximal w.r.t this property.

(i.e. aIM =Q⇒ a ¶ q)

Proof. (i)⇒ (ii)
Suppose, Q is a classical quasi primary element of M. Let, abr ¶ q, a, b, r ∈ L. This gives,

abr IM ¶ Q. By classical quasi primality of Q, we have, either (ab)k IM ¶ Q or rk IM ¶ Q

for some k ∈ Z+. i.e. (ab)k ¶ (Q : IM ) or rk ¶ (Q : IM ). As, Q is a classical quasi primary

element and IM 6¶ Q, (Q : IM ) is a quasi primary element of L i.e.
p

(Q : IM ) is a prime

element of L, by Theorem 5. Hence abr IM ¶ Q implies ab ¶
p

(Q : IM ) or r ¶
p

(Q : IM ).

Thus again, a ¶
p

(Q : IM ) or b ¶
p

Q : IM . So ak ¶ (Q : IM ) or bk ¶ (Q : IM ). Consequently,

akr ¶ (Q : IM ) or bkr ¶ (Q : IM ) i.e. akr ¶ q or bkr ¶ q. Hence, q is classical quasi primary.

(ii)⇒ (iii)
Suppose, q = (Q : IM ) is a classical quasi primary element of L. Since, M is a multiplication

module, Q = aIM for some a ∈ L. We have, qIM ¶ Q and a ¶ (Q : IM ) = q. Thus, aIM = Q

gives a ¶ q. So, q is maximal w.r.t. Q = qIM , (q ∈ L)

(iii)⇒ (i)
Suppose, Q = qIM where q is classical quasi primary element which is maximal w.r.t. this

property. So, aIM = Q implies a ¶ q. We show that, Q is a classical quasi primary element of

M. Let abN ¶ Q where a, b ∈ L, N ∈ M . Suppose, bkN 6¶ Q for any integer k. Since M is a



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 179

multiplication L-module. N = cIM for some c ∈ L. Thus, abcIM ¶ Q i.e. abc ¶ (Q : IM ) ¶ q

where q is a classical quasi primary element of L. Now, abc ¶ q and bkcIM 6¶Q implies

bkc 6¶ (Q : IM ) = q for any k. Therefore we have akc ¶ q for some integer k. So, akcIM ¶ qIM ,

i.e. akN ¶ qIM =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]. Fur-

ther 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 = Q1 ∧Q2 ∧ · · · ∧Qn be a primary decomposition

of Q with pi =
p

(Q i : IM ). If p1 ¶ p2 ¶ · · ·¶ pn then Q is a classical p1-quasi primary element.

Proof. Assume that, abN ¶Q, where a, b ∈ L, N 6¶Q. Then, N 6¶Q i , for some i (1¶ i ¶ n).

Suppose, t (1¶ t ¶ n) is the smallest number such that N 6¶Q t . Thus, N ¶Q1∧Q2∧· · ·∧Q t−1.

We have abN ¶ Q t and Q t is pt primary. Hence, (ab)k1 IM ¶ Q t for some k1 ∈ Z+ implies

ab ¶
p

(Q t : IM ) = pt . Thus, a ¶ pt or b ¶ pt . Now pt ¶ pt+1 ¶ · · · ¶ pn. Suppose,

a ¶ pt =
p

(Q t : IM ). Consequently ak IM ¶ Q t , for some integer k. If b ¶ pt =
p

(Q t : IM )

we have bk IM ¶ Q t . Thus ak IM ¶ Q t or bk IM ¶ Q t for some k ∈ Z+ where t is the smallest

positive integer such that N 6¶Q t . We have N 6¶Q t+1, . . . , N 6¶Qn, ak IM ¶Q t ,Q t+1, . . . ,Qn or

bk IM ¶Q t ,Q t+1, . . . ,Qn. Hence ak IM ¶Q t ∧Q t+1∧· · ·∧Qn or bk IM ¶Q t ∧Q t+1∧· · ·∧Qn for

some k ∈ Z+. But, N ¶ Q1 ∧Q2 ∧ · · · ∧Q t−1. It follows that, akN ¶ Q1 ∧Q2 ∧ · · · ∧Qn = Q or

bkN ¶Q1 ∧Q2 ∧ · · · ∧Qn. If N ¶Q, abN ¶Q implies aN ¶Q, bN ¶Q. Also,
p

(Q : IM ) =
p

(Q1 ∧Q2 ∧ · · · ∧Qn) : IM = p1. Therefore, Q is a classical p1- quasi primary

element.

Remark 1. The following example shows that the above theorem is not necessarily true if

Q1,Q2, . . . ,Qn are only assured to be classical (quasi) primary elements.

Example 5. Let R = Z , M = Z2 ⊕ Z3 ⊕ 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 Q1 = Z2 ⊕ (0)⊕ (0). Q2 = (0)⊕ Z3 ⊕ (0). Then Q1,

Q2, are elements of L(M). It can be shown that Q1 and Q2 are classical (quasi) primary elements

of L(M). Also, (0) =Q1 ∩Q2 and
p

(Q1 : M) =
p

(Q2 : M) = (0). Obviously,

2×3(Z2⊕Z3⊕(0)) = (0). Also, for each k ≥ 1,2k(Z2⊕Z3⊕(0)) 6⊆ (0) and 3k(Z2⊕Z3⊕(0)) 6⊆ (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 Q1 ∧
Q2 ∧ · · · ∧Qn 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 6¶Q then (Q : N) is a p-primary element.



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 180

Proof. First we show that (Q : N) is a primary element. Let a, b ∈ L, ab ¶ (Q : N) and

suppose a 6¶ (Q : N). Also as ab ¶ (Q : N), abN ¶Q with aN 6¶Q and Q is a primary element

implies that bn ¶ (Q : IM ) for some integer n. But bn IM ¶Q implies bnN ¶Q and we have

b ¶
p

(Q : N). Therefore, (Q : N) is a primary element of L. Now since N 6¶Q, there exists

A¶ IM and A¶ N such that A 6¶Q. Let a ¶
p

(Q : N). Then anN ¶Q and anA¶Q. But A 6¶Q

and Q is primary implies that (an)k = am ¶ (Q : IM ) for some integer m. That is

a ¶
p

(Q : IM ) = p and
p

(Q : N) ¶ p. Conversely, let a ¶
p

(Q : IM ) = p. Hence, an IM ¶ Q

for some integern. So anN ¶ Q for some integer n. Thus an ¶ (Q : N) and a ¶
p

(Q : N).

This shows that p ¶
p

(Q : N) and we have
p

(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 6= IM be an element of a lattice module M and assume that N has a primary

decomposition. Let N = Q1 ∧Q2 ∧ · · · ∧Qk be a reduced primary decomposition of N and p be

prime element of L. Then following statements are equivalent,

(i) p =
p

Q i for some i

(ii) (N : X ) is a p-primary element of L for some X 6¶ N.

Proof. (i)⇒ (ii)
Let N = Q1 ∧ Q2 ∧ · · · ∧ Qk be a reduced primary decomposition of N. First suppose that,

p =
p

Q i for some i. Without loss of generality we can assume that p =
p

(Q1 : IM ) where

pi =
p

(Q i : IM ) i = 1,2, . . . , k. We prove that, (N : X ) is a p-primary element of L for some

X 6¶ N . Since the decomposition is reduced Q i � Q1 ∧ Q2 ∧ · · · ∧ Q i−1 ∧ Q i+1 ∧ · · · ∧ Qk for

i = 1,2, . . . , k. In particular, Q1 � Q2 ∧Q3 ∧ · · · ∧Qk. So there exists X ¶ Q2 ∧Q3 ∧ · · · ∧Qk

such that X 6¶Q1 and hence X 6¶ N =Q1 ∧Q2 ∧ · · · ∧Qk. Also

(N : X ) = (Q1 ∧Q2 ∧ · · · ∧Qk) : X = (Q1 : X )∧ (Q2 : X )∧ · · · ∧ (Qk : X ).

For i = 2,3, . . . , k we show that (Q i : X ) = 1. Since X ¶ Q2 ∧Q3 · · · ∧Qk, 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 ) ¶ 1 implies (Q i : X ) = 1 for i = 2,3, . . . , k.

Hence, (N : X ) = (Q1 : X )∧1∧· · ·∧1= (Q1 : X ). So by the above result, (Q1 : X ) is p-primary

element implies (N : X ) is a p-primary element of L where X 6¶ N .

(ii)⇒ (i)
Assume that (N : X ) is a p-primary element of L for some X 6¶ N , X ∈ M . We prove that
p

Q i = p for some i. We have,

p =
Æ

(N : X ) =
Æ

[(Q1 ∧Q2 ∧ · · · ∧Qk) : X ] =
Æ

(Q1 : X )∧
Æ

(Q2 : X )∧ · · · ∧
Æ

(Qk : X ).

We claim that for each i,
p

(Q i : X ) = pi or 1 and equal to pi for at least one i. We have,

X 6¶ N = Q1 ∧Q2 ∧ · · · ∧Qk implies X 6¶ Q i for at least one i (1 ¶ i ¶ k). Suppose, X 6¶ Qr

(1 ¶ r ¶ k) and X ¶ Q1 ∧ Q2 ∧ · · · ∧ Qr−1 ∧ Qr+1 ∧ . . .Qk that is X ¶ ∧Q i , where (i 6= r).



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 181

Let a ¶ (Q i : X ), a ∈ L. Since, X ¶ ∧Q i . We have, aX ¶ Q i for all i 6= r and a ∈ L. Hence,

a ¶
p

(Q i : X ) for all a ∈ L. In particular, 1 ¶
p

(Q i : X ) for all i 6= r. But,
p

(Q i : X ) ¶ 1 for

all i 6= r. Therefore,
p

(Q i : X ) = 1 for all i 6= r. For i = r, X 6¶ Qr . Let a ¶
p

(Qr : X ). Hence,

anX ¶Qr , for some positive integer n, where X 6¶Qr . As Qr is primary,

an ¶
p

(Qr : IM ) = pr . Thus, a ¶ pr , since pr is prime and we have,
p

(Qr : X ) ¶ pr . On the

other hand, let a ¶ pr =
p

Qr =
p

(Qr : IM ). Hence, an ¶ (Qr : IM ) for some positive integer

n. That is an IM ¶ Qr and therefore, anX ¶ Qr , for some positive integer n. Consequently,

an ¶ (Qr : X ) and hence a ¶
p

(Qr : X ). This gives pr ¶
p

(Qr : X ). Hence,
p

(Qr : X ) = pr

where X 6¶ Qr . We have shown that for each i,
p

(Q i : X ) = pi or 1 and is equal to pi for at

least one i, since X 6¶ N . Then, p =
p

(N : X ) =
p

(Q1 : X ) ∧ · · · ∧p(Qk : X ) is the meet of

some of the prime elements p1, p2, . . . , pl (1¶ l ¶ k). That is p =
p

(N : X ) = p1∧p2∧· · ·∧ pl .

We show that p = pi for some i. We have, p ¶ pi i = 1,2, . . . , l. If for each i p 6= pi then pi 6¶ p

for all i = 1,2, . . . , l. This implies that there exist x i ¶ pi such that x i 6¶ p for all i = 1,2, . . . , l.

Then, x1 x2 . . . x l ¶ p1 ∧ p2 ∧ · · · ∧ pl = p. This shows that x i ¶ p for at least one i (1 ¶ i ¶ k)

a contradiction. Hence, p = pi 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 =Q1∧Q2∧· · ·∧Qn with

pi =
p

(Q i : IM ) be a reduced primary decomposition of Q. Then Q is a classical quasi primary

element if and only if {p1, p2, . . . , pn} is a chain of prime elements of L. In that case, radical of

(Q : IM ) is the smallest of the primes p1, p2, . . . , pn.

Proof. Since, Q =Q1 ∧Q2 ∧ · · · ∧Qn is a reduced primary decomposition of Q by Theorem

7 for each i (1 ¶ i ¶ n), pi =
p

(Q : Ai) for some Ai 6¶ Q where pi =
p

(Q i : IM ). Assume

that, Q is a classical quasi primary element. We show that {p1, p2, . . . , pn} is the chain of prime

elements of L. Assume that, it is not a chain. Then pi 6¶ p j and p j 6¶ pi for some i 6= j. Select

a ¶ pi such that a 6¶ p j and b ¶ p j such that b 6¶ pi . i.e. a ¶
p

(Q i : IM ), a 6¶ Æ(Q j : IM )

and b ¶
Æ

(Q j : IM ), b 6¶p(Q i : IM ). Since pi =
p

(Q : Ai), Ai 6¶Q for each i = 1,2, . . . , n, we

have, a ¶
p

(Q i : IM ) =
p

(Q : Ai) and a 6¶Æ(Q : A j). Similarly, b ¶
Æ

(Q j : IM ) =
Æ

(Q : A j)

and b 6¶ p(Q : Ai). This shows that aki Ai ¶ Q for some integer ki and akA j 6¶ Q for any k.

Similarly, bk j A j ¶ Q for some integer k j and bkAi 6¶ Q for any k. Thus aki Ai ∨ bk j A j ¶ Q and

akA j ∨ bkAi 6¶ Q for any k. Let k = max{ki , k j}. Then ak bkAi ∨ bkakA j = ak bk(A j ∨ Ai) ¶

Q. Since, Q is a classical quasi primary element of M, it follows that (ak)l(Ai ∨ A j) ¶ Q or

(bk)l(Ai ∨ A j) ¶ Q i.e. as(Ai ∨ A j) ¶ Q or bs(Ai ∨ A j) ¶ Q for some integer s. Therefore,

as ¶ (Q : A j) or bs ¶ (Q : Ai) for some integer s. Hence, a ¶ p j or b ¶ pi , which is a

contradiction. Thus, {p1, p2, . . . , pn} 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 =Q1 ∧Q2 ∧ · · · ∧Qn 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



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 182

Q1 = 2Z ⊕ Z, Q2 = Z ⊕ 3Z, Q3 = Z ⊕ (0), Q4 = (0)⊕ Z. Then, Q1, Q2, Q3, Q4 are primary sub

modules of M and hence, these are the elements of a lattice module L(M) with
p

(Q i : M) = 2Z,
p

(Q2 : M) = 3Z and
p

(Q i : M) =
p

(Q4 : M) = (0). Also, (0) =Q1∩Q2∩Q3∩Q4 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 (respec-

tively 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 i t is classical primary (respectively classical quasi primary) element

where (i1, i2, . . . , it) ⊆ {1,2, . . . , n} for t ¾ 2 with i1 < i2 < · · ·< it

(ii) for each j, Q j ∧
i 6= j

Q i

Corresponding to the above definition by Theorem 2 we have a list of prime elements
p

(Q1 : IM ), . . .
p

(Qn : IM ). 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 =Q1∧Q2∧· · ·∧Qk 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 pi =
p

(Q i : IM ). Then p1, p2, . . . , pk are the prime elements which are called

prime divisors of N or associated primes of N. We have

Æ

(N : IM ) =
Æ

(Q1 ∧Q2 ∧ · · · ∧Qk) : IM

=
Æ

(Q1 : IM )∧
Æ

(Q2 : IM )∧ · · · ∧
Æ

(Qk : IM )

=p1 ∧ p2 ∧ · · · ∧ pk.

If a ¶ p1p2p3 . . . pk then a ¶ pi for each i = 1,2, . . . , n. Hence a ¶
p

(Q i : IM ) for each i. That

is a ¶
p

(Q1 : IM ) ∧
p

(Q2 : IM ) ∧ . . .
p

(Qk : IM ) =
p

(N : IM ). So an ¶ (N : IM ) for some

positive integer n. If p is any prime element of L containing (N : IM ) then an ¶ p and hence

a ¶ p. Then p1p2 . . . pk ¶ p whenever (N : IM )¶ p. This shows that pi ¶ p for some i. If p is a

minimal prime element containing (N : IM ) then pi ¶ p for some i. Also pi contains (N : IM ),

but p is minimal such element. Hence pi = 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.



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 8 (2015), 172-184 183

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 = Q1 ∧Q2 ∧ · · · ∧Qn with
p

(Q i : IM ) = pi for i = 1,2, . . . , n be a reduced classical

quasi primary decomposition of N. Then, {pi | i = 1,2, . . . , n}= min(N : IM )} and the set

{pi | i = 1,2, . . . , n} is uniquely determined.

Proof. First we show that, min(N : IM ) ⊆ {pi | i = 1,2,3, . . . , n}. Let p be a minimal prime

of (N : IM ). Then p is a minimal member of associated primes of N. But p = pi for some i if

and only if there exists A 6¶ N in M such that (N : A) p-primary. Thus,

p =
p

(N : A) =
p

(Q : IM ) for some A 6¶ N . Renumber the Q i such that A 6¶ Q i for 1 ¶ i ¶ j

and A ¶ Q i for j + 1 ¶ i ¶ n. Since pi =
p

(Q i : IM ), pi
ki IM ¶ Q i for some integer ki ,

(1 ¶ i ¶ n). Therefore, (
j∧

i=1
pi

ki )A¶
n∧

i=1
Q i = N and so

j∧
i=1

pi
ki ¶ (N : A) ¶ p. Since p is prime,

pt ¶ p for some t ¶ j. As, (N : IM ) ¶
p

(N : IM ) ¶
p

(Q t : IM ) = pt and p is a minimal prime

of (N : IM ), we conclude that p = pt . Now it is sufficient to show that each pi(1 ¶ i ¶ n) is a

minimal prime of (N : IM ). Without loss of generality, we may take i = 1. Clearly

(N : IM )¶
Æ

(N : IM ) =
Æ

(Q1 ∧Q2 ∧ · · · ∧Qn : M) =
n∧

i=1

Æ

(Q i : IM )¶ p1.

On the contrary, suppose that p1 is not a minimal prime of (N : IM ). Thus ∃ an i ∈ {1,2, . . . , n}
such that pi is minimal prime of (N : IM ) with pi < p1 (since

min(N : IM ) ⊆ {pi | i = 1,2,3, . . . , n}). Again without loss of generality, we may take i = 2.

Thus (N : IM ) ¶ p2 < p1. By [5], each Q i has a minimal primary decomposition. Suppose

that, Q1 = Q11 ∧ · · · ∧Q1s with
Æ

(Q1 j : IM ) = p1 j (1 ¶ j ¶ s) and Q2 = Q21 ∧ · · · ∧Q2t with
Æ

(Q2 j : IM ) = p2 j (1¶ j ¶ t) are a reduced primary decompositions of Q1 and Q2 respectively.

By Theorem 8 {p1 j | 1 ¶ j ¶ s} and {p2 j | 1 ¶ j ¶ t} are chains of prime elements. Without

loss of generality, we can assume that, p11 ⊆ p12 ⊆ · · · ⊆ p1s and p21 ⊆ p22 ⊆ · · · ⊆ p2t . We

thus get p1 = p11 and p2 = p21, since

p1 =
Æ

(Q1 : IM ) =
Æ

(Q11 ∧ · · · ∧Q1s) : IM =
s∧

i=1

Æ

(Q1i : IM ) = p11

and

p2 =
Æ

(Q2 : IM ) =
Æ

(Q21 ∧ · · · ∧Q2t) : IM =
t∧

i=1

Æ

(Q2i : IM ) = p21.

It follows that, p21 ⊆ p11 ⊆ p12 ⊆ · · · ⊆ p1s and so by Theorem 8,

Q1

′
=Q21∧Q11∧· · ·∧Q1s is a classical quasi primary element of M with

q

(Q1

′
: IM ) = p21 = p2.

On the other hand,

N =Q1 ∧ · · · ∧Qn = (Q11 ∧ · · · ∧Q1s)∧ (Q21 ∧ · · · ∧Q2t)∧Q3 · · · ∧Qn

=(Q21 ∧Q11 ∧ · · · ∧Q1s)∧ (Q21 ∧ · · · ∧Q2t)∧ (Q3 ∧ · · · ∧Qn).

Thus, N =Q
′
1 ∧Q2 ∧ · · · ∧Qn is a classical quasi primary decomposition of N with

q

(Q
′
1

: IM ) =
p

(Q2 : IM ) = p2 and
p

(Q i : IM ) = pi for i = 3, . . . n. We note that if ∃ another

Q i (3 ¶ i ¶ n) such that
p

(Q i : IM ) = pi = p1 then by similar arguments we can replace it



REFERENCES 184

by Q
′
i

such that
q

(Q
′
i
: IM ) =
p

(Q2 : IM ) = p2. Now, by using this decomposition we can

obtain a reduced classical quasi primary decomposition, N = Q
′′
1 ∧ Q

′′
2 ∧ · · · ∧ Q

′′
k

such that

p1 /∈ {
q

(Q
′′
i

: IM ) | i = 1,2,3,4, . . . , k} ⊆ {pi | i = 2, . . . , n}, contrary with the irrenduntness of

the decomposition N =Q1,∧Q2 ∧ . . . ,∧Qn with

{
Æ

(Q i : IM ) | i = 1,2,3, . . . , n}= {pi | i = 1,2 . . . , n}.

Thus, {pi | i = 1,2, . . . , n}= min(N : IM ).

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.

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