main.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 7, No. 2, 2014, 201-209

ISSN 1307-5543 – www.ejpam.com

Primary Decomposition in Lattice Modules

C S Manjarekar 1, U N Kandale2,∗

1 Department of Mathematics, Shivaji University, Kolhapur, Maharashtra, India
2 General Engineering Department, Sharad Institute of Technology College of Engineering, Shivaji

University, Kolhapur, India

Abstract. In this paper, we study primary decomposition of elements in lattice modules. A neces-

sary and sufficient condition for a prime element p of a multiplicative lattice L to be equal to some

associated prime of an element in a lattice module having primary decomposition is obtained.

2010 Mathematics Subject Classifications: 13A99

Key Words and Phrases: Prime element,primary element,lattice modules,primary decomposition

1. Introduction

A multiplicative lattice L is a complete lattice provided with commutative, associative and

join distributive multiplication in which the largest element 1 acts as a multiplicative identity.

An element a ∈ L is called proper if a < 1. A proper element p of L is said to be prime if

ab ≤ p implies a ≤ p or b ≤ p. If a ∈ L,b ∈ L, (a : b) is the join of all elements c in L such that

cb ≤ a. A proper element p of L is said to be primary if ab ≤ p implies a ≤ p or bn ≤ p for

some positive integer n. If a ∈ L, the radical of a denoted by
p

a = ∨{x ∈ L | xn ¶ a, n ∈ Z+}.
An element a ∈ L is called compact if a ¶ ∨

α

b
α

implies a ¶ b
α1 ∨ b

α2 ∨ . . . ∨ b
αn for some

finite subset {α1,α2, . . . ,αn}. Throughout this paper, L denotes a multiplicative lattice which

satisfies the ACC so that each element of L is compact. If q is a primary element of L then

pq = ∨{x ∈ L | xn ¶ q, for some integer n}

is a prime element containing q. It is easily verified that, pq is a minimal prime containing

q [2]. The prime pq which is same as
p

q is called the prime associated with q and has the

properties, pk
q ¶ q ¶ pq for some integer k and ab ¶ q implies a ¶ q or b ¶ pq.

An element a is said to have a primary decomposition if there exist primary elements

q1,q2, . . . ,qn such that a = q1 ∧ q2 ∧ . . . ∧ qn. If some qi contains the meet of remaining ones

∗Corresponding author.

Email addresses: 
smanjrekar�yahoo.
o.in (C. Manjarekar), ujwalabiraje�gmail.
om (U. Kandale)

http://www.ejpam.com 201 c© 2014 EJPAM All rights reserved.



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 7 (2014), 201-209 202

then this qi can be dropped from the primary decomposition. After deleting such primary com-

ponents and combining the primary components with same associated prime we get a reduced

primary decomposition in which distinct primaries are associated with distinct primes.Such a

primary decomposition is also called an irredundant primary decomposition, reduced primary

decomposition or normal primary decomposition. Let a = q1∧q2 . . .∧qn be a reduced primary

decomposition of a and let p1, p2, . . . , pn denotes the associated primes of q1,q2, . . . ,qn re-

spectively,which are also called associated primes of a. A subset C of {p1, p2, . . . , pn} is called

isolated if pi ∈ C implies p j ∈ C whenever p j ¶ pi .

Let M be a complete lattice and L be a multiplicative lattice. Then M is called L-module or

module over L if there is a multiplication between elements of L and M written as aB where

a ∈ L and B ∈ M which satisfies the following properties,

i) (∨
α

a
α
)A= ∨

α

a
α
A ∀a

α
∈ L, A∈ M

ii) a(∨
α

A
α
) = ∨

α

aA
α
∀a ∈ L, A

α
∈ M

iii) (ab)A= a(bA) ∀a, b ∈ L, A∈ M

iv) IB = B

v) 0B = 0M , for all a, a
α
, b ∈ L and A,A

α
∈ M ,

where I is the supremum of L and 0 is the infimum of L. We denote by 0M and IM the

least element and the greatest element of M . The elements of L will generally be denoted by

a, b, c, . . . and elements of M will generally be denoted by A, B, C . . ..

Let M be a L-module. If N ∈ M and a ∈ L then (N : a) = ∨{X ∈ M | aX ¶ N}. If

A, B ∈ M , then (A : B) = ∨{x ∈ L | xB ¶ A}. An L-module M is called a multiplication

L-module if for every element N ∈ M there exists an element a ∈ L such that N = aIM [4].

A proper element N of M is said to be prime if aX ¶ N implies X ¶ N or aIM ¶ N that

is a ¶ (N : IM ) for every a ∈ L, X ∈ M . An element N < IM in M is said to be primary if

aX ¶ N implies X ¶ N or an IM ¶ N that is an ¶ (N : IM ) for some integer n. An element

N of M is called a radical element if (N : IM ) =
p

(N : IM ). Noether lattice is a modular

multiplicative lattice satisfying ascending chain condition in which every element is the join

of principal elements. Let N be an element of a lattice module M . Then N is said to have a

primary decomposition if there exist primary element Q1,Q2, . . . ,Qn such that

N = Q1 ∧Q2 ∧ . . .Qn. 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 components which

contains the meet of remaining ones can be dropped from the primary decomposition. If no

Q i can be dropped further we get a reduced primary decomposition of N . Such a primary

decomposition is also called an irredundant primary decomposition. If Q is primary then
p

Q =
p

(Q : IM ) is prime. We note that,
p

(N : IM ) may also be denoted by
p

N .

2. Primary Decomposition of Elements

The following result gives the relation between a primary element Q and
p

(Q : IM ).



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 7 (2014), 201-209 203

Theorem 1. If Q is a primary element of a lattice module M then
p

(Q : IM ) is a prime element

of L. If a is an element of L and if a ¶ p where p is a prime element of L then
p

a ¶ p.

Proof. Let ab ¶
p

(Q : IM ) and suppose b 

p

(Q : IM ). Then (ab)n = an bn ¶ (Q : IM ) for

some positive integer n. Now b 

p

(Q : IM ) implies bm 
 (Q : IM ) for any positive integer m.

In particular bn 
 (Q : IM ). As Q is primary, (an)k ¶ (Q : IM ) for some positive integer k. That

is at ¶ (Q : IM ) for some positive integer t and a ¶
p

(Q : IM ). Therefore
p

(Q : IM ) is prime.

Let a ¶ p. Take any x ¶
p

a. Then xn ¶ a ¶ p for some positive integer n. As p is prime,

x ¶ p and hence
p

a ¶ p.

The following theorem gives the relation between meet of primary elements and their

equal associated primes.

Theorem 2. If Q1,Q2, . . . ,Qk are p-primary elements of a lattice module M then Q1∧Q2∧. . .∧Qk

is p-primary.

Proof. By hypothesis
p

(Q i : IM ) = p, i = 1,2, . . . , k. Let Q = Q1 ∧Q2 ∧ . . . ∧Qk. We have,
p

∧Q i =
p

((∧Q i) : IM ) =
p

(Q1 : IM ) ∧
p

(Q2 : IM ) ∧ . . . ∧
p

(Qk : IM ) = p. We show that

∧Q i is primary, where i = 1,2, . . . , k. Let aX ¶ Q = ∧Q i where a ∈ L, X ∈ M . Suppose,

X 
 Q. Then X 
 Q i for some i (1 ¶ i ¶ k). As Q i is primary, aX ¶ Q i and X 
 Q i implies

a ¶
p

(Q i : IM ) = p. That is a ¶
p

(Q : IM ). Therefore, Q is primary.

It is shown by Thakare and Manjarekar [6] that the radical of any element a of a multi-

plicative lattice satisfing the ACC can be written as the meet of minimal prime divisors of a.

Hence, we have the following result.

Theorem 3. Let L be a multiplicative lattice satisfing the ACC and N be an element of M then
p

(N : IM ) = ∧{p | p is minimal prime containing (N : IM )}.
An element N of a lattice module M is said to be meet irreducible if for any two elements

A1 and A2 of M , N = A1 ∧ A2 implies either A1 = N or A2 = N .

Theorem 4. If a lattice module M satisfies ACC the chain A1 ¶ A2 ¶ . . . implies there exist

positive integer m such that An = Am for all n ≥ m. Then every element of M can be written as

the meet of a finite number of meet irreducible elements of M.

Proof. Let τ be the set of all elements of M which can not be written as a meet of a

finite number of meet irreducible elements of M . If τ is empty we have nothing to prove.

Suppose τ is not empty. As M satifies ACC, τ has a maximal element say N . As N ∈ τ, N

is not irreducible. So there exists elements A1 and A2 of M such that N = A1 ∧ A2 where

N 6= A1, N 6= A2. So, N < A1, N < A2. This shows that both A1 and A2 can be written as the

meet of a finite number of meet irreducible elements of M . So there are irreducible elements

K1, K2, . . . , Km and K
′
1, K

′
2, . . . , K

′
n of M such that A1 = K1 ∧ K2 ∧ . . .∧ Km and

A2 = K
′
1 ∧K

′
2 ∧ . . .∧ K

′
n. But then N = K1 ∧K2 . . .∧ Km ∧K

′
1 ∧ K

′
2 ∧ . . . K

′
n. That is N is the meet

of a finite number of meet irreducible elements. This contradicts the fact that N ∈ τ. Hence,

τ is empty.



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 7 (2014), 201-209 204

The study of primary elements and their associated primes for modules is carried out by

P J Mc Carthy and Larsen [5]. We give eqivalent formulation in the next theorems for lattice

modules.

Theorem 5. Let Q be a p-primary element of lattice module M and N be an element of M. If

N 
 Q then (Q : N) is a p-primary element.

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

suppose, a 
 (Q : N). As a 
 (Q : N), aN 
 Q. Also as ab ¶ (Q : N), abN ¶ Q. But aN 
 Q

and Q is a primary element implies that bn ¶ (Q : IM ) for some integer n. But bn IM ¶ Q

implies bnN ¶ Q. Hence, b ¶
p

(Q : N). Therefore, (Q : N) is a primary element of L. Now

since N 
 Q, there exists A∈ M and A¶ N such that A
 Q. Let a ¶
p

(Q : N). Then anN ¶ Q.

Hence, anA ¶ Q. But A 
 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, anIM ¶ Q for some integer n. 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.

Theorem 6. Let M be a lattice module and a be an element of L, p be a prime element of L and

Q be p-primary element of M. If a 
 p then (Q : a) = Q.

Proof. Suppose a 
 p where p =
p

(Q : IM ). Since, a 
 p there is some b ¶ a such that

b 
 p. Let X ¶ (Q : a). Then aX ¶ Q and hence bX ¶ Q where b 

p

(Q : IM ) = p. As Q

is a p-primary, X ¶ Q. Hence, (Q : a) ¶ Q. Conversely let X ¶ Q. Since a ¶ 1, aX ¶ Q. So

X ¶ (Q : a) and hence Q ¶ (Q : a). Therefore,Q = (Q : a).

The following theorem gives the characterisation of a prime element p of L to be equal to

some associated prime of an element which has a primary decomposition.

Theorem 7. 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 p =
p

Q i for some i if and only if (N : X ) is a p-primary element of L

for some X 
 N.

Proof. 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 
 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 
 Q1 and hence X 
 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



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 7 (2014), 201-209 205

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 above result, (Q1 : X ) is p-primary element

implies (N : X ) is a p-primary element of L where X 
 N . Conversely assume that (N : X ) is a

p-primary element of L for some X 
 N , X ∈ M . We prove that
p

Q i = p for some i. We have,

p =
p

(N : X ) =
p

[(Q1 ∧Q2 ∧ . . . ∧Qk) : X ] =
p

(Q1 : X )∧
p

(Q2 : X )∧ . . . ∧
p

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

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

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 
 Qr . Let a ¶
p

(Qr : X ). Hence,anX ¶ Qr , for some positive integer

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

This leads us to the following result.

Theorem 8. Let N 6= IM be an element of a lattice module M and assume that N has a primary

decomposition. If N = Q1 ∧Q2 ∧ . . . ∧Qm = S1 ∧ S2 ∧ . . . ∧ Sn are two reduced primary decom-

positions of N then n = m and the Q i and Si can be so numbered that
p

(Q i : IM ) =
p

(Si : IM )

for i = 1,2, . . . , n.

The above theorem proves the uniqueness of associated primes in reduced primary decom-

position. The next result gives the relation between zero divisors of L and associated primes

of zero.

Theorem 9. Let L be a Noetherian lattice and p1, p2, . . . , pk be the prime divisors of the element

0 that is associated prime elements of element 0. Then every zero divisors of L is contained in

p1 ∨ p2 ∨ . . .∨ pk.



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 7 (2014), 201-209 206

Proof. Let 0 = q1 ∧ q2 ∧ . . . ∧ qk be a reduced primary decomposition of 0 and pi =
p

qi,

i = 1,2, . . . , k. Suppose a is a zero divisor. Then if a = 0 obviously a ¶ p1 ∨ p2 ∨ . . . ∨ pk.

Suppose, a is a proper zero divisor that is a 6= 0 and let ab = 0 where b 6= 0. Now,

ab = 0¶ q1∧q2∧ . . .∧qk = {0}. Hence, ab ¶ qi for all i and b 
 qi for at least one i. Because,

b ¶ qi for all i implies b ¶ q1 ∧ q2 ∧ . . . ∧ qk = {0} and hence b = 0 , a contradiction. Let

b 
 q j . Then, ab ¶ q j, b 
 q j and q j is a primary element. Therefore, a ¶
p

q j = p j, which

shows that a ¶ p1 ∨ p2 ∨ . . .∨ pk.

Theorem 10. Let M be a lattice module and N 6= IM be an element of M which has a reduced

primary decomposition N = Q1 ∧Q2 ∧ . . . ∧Qm. If every Q i (1¶ i ¶ m) is a prime element then

(N : IM ) =
p

(N : IM ) and the converse holds if (Q i : IM ) are prime elements.

Proof. Suppose each Q i is a prime element. Let a ¶
p

(N : IM ). Then

an IM ¶ N = Q1 ∧Q2 ∧ . . .∧Qm

for some positive integer n. This implies that, an IM ¶ Q i for each i. As Q i is a prime element,

aIM ¶ Q i or an−1 IM ¶ Q i . If aIM ¶ Q i then a ¶ (Q i : IM ). Otherwise an−1 IM ¶ Q i implies

aIM ¶ Q i or an−2 IM ¶ Q i. Continuing in this way we obtain, a ¶ (Q i : IM ) for each i.

Therefore a ¶ (Q1 : IM )∧ (Q2 : IM )∧ . . . ∧ (Qm : IM ). That is a ¶ (N : IM ) and hence

(N : IM ) =
p

(N : IM ). Conversely assume that, (N : IM ) =
p

(N : IM ). We show that

(Q i : IM ) = pi. Let y ¶ pi =
p

(Q i : IM ). As
m∧

i=1
pi is irredundant(reduced) there exists z ¶ ∧p j

such that z � pi in L. Now yz ¶
m∧

i=1
pi =

m∧
i=1

p

(Q i : IM ) =
m∧

i=1
(Q i : IM ) implies yzIM ¶ Q i for

each i. Since Q i is primary, z � pi gives y ¶ (Q i : IM ). Hence, pi ¶ (Q i : IM ). Consequently,

pi = (Q i : IM ).

Now we obtain a characterization of a prime element p of L containing some associated

prime pi of N 6= IM in a lattice module M .

Theorem 11. Let N 6= IM have a reduced primary decomposition N = Q1 ∧Q2 ∧ . . . ∧Qm and

pi =
p

(Q i : Im) be the associated primes of N. For a prime element p of L to contain (N : IM ) it

is necessary and sufficient that p contains pi for some i.

Proof. Suppose pi ¶ p for some i. Then (N : IM ) =
m∧

i=1
(Q i : IM ) implies (N : IM ) ¶ p.

Conversely assume that (N : IM )¶ p. Then (Q1 : IM )∧ (Q2 : IM )∧ . . . ∧ (Qm : IM )¶ p implies

(Q i : IM ) ¶ p for some i. But
p

(Q i : IM ) = pi is the smallest prime containing (Q i : IM ).

Hence, pi ¶ p for some i.

In our next result we show that those Q
′s
i

can be uniquely determined which are isolated

primary components of N 6= IM .

Theorem 12. Let N 6= IM have a reduced primary decomposition N = Q1 ∧Q2 ∧ . . . ∧Qm and

p1, p2, . . . , pm be the associated primes of Q1,Q2, . . . ,Qm respectively. The element

Q
′
i = ∨{X ∈ M | (N : X )� pi}



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 7 (2014), 201-209 207

is an element of M which is contained in Q i . If Q i is an isolated primary component of N then

Q i = Q
′
i .

Proof. Take any element A ∈ {X ∈ M | (N : X ) � pi}. Then (N : A) � pi . So there exists

a ∈ L such that aA¶ N and a � pi =
p

(Q i : IM ). Hence, an IM � Q i for any integer n. Now

aA¶ Q i, anIM � Q i and Q i is primary gives A¶Q i . Hence Q i

′
¶ Q i and the first part is proved.

If pi is a minimal associated primes of N it follows that p j � pi for i 6= j. Then there exists

b j ¶ p j in L such that b j � pi. We have b j ¶ p j =
p

(Q j : IM ) = ∨{a j ∈ L | as j

j
IM ¶ Q j for some

integer s j}. Since each element of L is compact, we have b j ¶ p j =
n∨

j=1
{a j | as j

j
IM ¶ Q j for

some integer s j}. Put s1+ s2+ . . .+ sn = k( j). Then b j
k( j)IM ¶ (a1 ∨ a2 ∨ . . .∨ an)

k( j)IM ¶ Q j.

Clearly b = Π
j 6=i

b j
k( j) � pi as pi is prime. However, bIM ¶ ∧

j 6=i
Q j. Next take any X ¶ Q i. Then

X bIM ¶
m∧

i=1
Q i = N . So b ¶ (N : X )� pi. This implies that X ∈ {X ∈ M | (N : X ) 6= pi}. Hence

X ¶ ∨{X ∈ M | (N : X )� pi}= Q i

′
and we have Q i ¶ Q i

′
. Consequently, Q i = Q i

′
.

We now relate the radical of N with the isolated primes of N ∈ M . In that direction we

have:

Theorem 13. Let M be a lattice module and N 6= IM have an irredudent(reduced) primary

decomposition N = Q1 ∧Q2 ∧ . . . ∧Qn then
p

(N : IM ) is the meet of isolated prime elements of

N.

Proof. We have

p

(N : IM ) =
p

(Q1 ∧Q2 ∧ . . .Qn) : IM =
p

(Q1 : IM )∧
p

(Q2 : IM ) . . . ∧
p

(Qn : IM )

=p1 ∧ p2 ∧ . . .∧ pn,

where pi =
p

(Q i : IM ) are associated primes of N . If some pk is not isolated then pk ≥ pi for

some pi and hence we can delete such elements from the above primary decomposition and

we are through.

We note that an element A= aIM of M where a ∈ L is said to be nilpotent if an IM = 0M

for some positive integer n. If a lattice module M satisfies the ACC and if every element of

L is the join of meet prncipal elements then every element of M can be written as a meet of

finite number of primary elements [1].

Theorem 14. Let M be a lattice module satisfying the ACC over a multiplicative lattice L in

which every element is the join of meet principal elements. Then the join of the set of all elements

a ∈ L such that aIM is nilpotent is the meet of the isolated primes of 0M .

Proof. Let 0M =
n∧

i=1
Q i be a reduced primary decomposition of 0M and pi =

p

(Q i : IM ) be

an associated prime of Q i, i = 1,2, . . . n. We have

p

(0M : IM ) = ∨{a ∈ L | anIM = 0M for some positive integer n}



C. Manjarekar, U. Kandale / Eur. J. Pure Appl. Math, 7 (2014), 201-209 208

and
p

(0M : IM ) = p1∧ p2∧ . . .∧ pk where p1, p2, . . . , pk are the isolated primes of 0M . Hence,
p

(0M : IM ) is the meet of isolated primes of 0M .

The primeness of the radical of A∈ M is characterized in the following result.

Theorem 15. For A∈ M,
p

(A : IM ) is prime if and only if A has a single isolated prime element.

Proof. Let A = Q1 ∧ Q2 ∧ . . . ∧Qn be primary decomposition of A. If A has a single iso-

lated prime element p then
p

(A : IM ) = p. Conversely, assume that
p

(A : IM ) is prime and
p

(A : IM ) = p1 ∧ p2 where p1, p2 are isolated primes of A. Then there are x , y ∈ L such that

x ¶ p1, x � p2 and y ¶ p2, y � p1. Hence x y ¶
p

(A : IM ) which is prime. But then x ¶ p2

or y ¶ p1 which is a contradiction. Thus A has a single isolated prime element.

In the remaining part we assume that a lattice module M satisfies the ACC over a multi-

plicative lattice L in which every element is the join of meet principal elements. This condition

assures that any element M has a reduced primary decomposition.

Theorem 16. Let A be any element of M and b ∈ L be such that A 6= IM . Then A= (A : b) if and

only if b is contained in no associated prime element of A.

Proof. Let A = Q1 ∧Q2 ∧ . . . ∧Qm be an irredundant primary decomposition of A and let

pi =
p

(Q i : IM ). Suppose b � pi for any i = 1,2, . . . , m. This leads us to the fact bn � pi for

any positive integer n. We know that (A : b)b ¶ A [3] and thus (A : b)b ¶ Q i for all i. Since

Q i is primary and b � pi we have (A : b) ¶ Q i . That is (A : b) ¶
m∧

i=1
Q i = A. But A ¶ (A : b)

gives (A : b) = A. Conversely, suppose (A : b) = A and if possible without loss of generality

assume that b ¶ p1. Then (Q : bs) = IM for some integer s. We have (A : b) : b = A : b2 [3].

Continuing in this way A : b = A : bs. But A : b = A implies A : bs = A. Finally,

A=(A : bs) = ((Q1 ∧Q2 ∧ . . .∧Qm) : bs) = ((Q1 : bs)∧ (Q2 : bs)∧ . . . ∧ (Qm : bs)

= ∧
j 6=1
(Q j : bs) ≥ ∧

j 6=1
Q j ≥ A.

That is A= ∧
j 6=1

Q j. This contradicts the fact that A=
m∧

i=1
Q i is a reduced primary decomposition

of A.

The above theorem can be restated in the following form.

Theorem 17. Let N 6= IM have a reduced primary decomposition Q1 ∧ Q2 ∧ . . . ∧ Qm and

p1, p2, . . . , pm be the associated primes of Q
′s
i
. For an element b of L to be contained in some

associated prime element of N it is necessary and sufficient that (N : b) 6= N.

Direct application of the above theorem gives the following result.

Theorem 18. For an element b of L to be contained in some associated prime element of N, it is

necessary and sufficient that there is an element Y � N such that bY ¶ N.



REFERENCES 209

An element X ∈ M is called a zero divisor if (0M : X ) 6= 0 so there exists a 6= 0 in L such

that aX = 0M .

Theorem 19. Let M be a lattice module where M satisfies the ACC and every element of M is the

join of meet principal elements. If X ∈ M then the join of all a ∈ L such that a 6= 0 and aX = 0M

is contained in the join of all associated prime elements of 0M .

Proof. Let X be a zero divisor of M . Then 0M : X 6= 0 that is there exists a 6= 0 in L such

that aX = 0M . We know that for an element b of L (b 6= 0, bX = 0M ) to be contained in some

associated prime of 0M it is necessary and sufficient that

(0M : b) = ∨{X ∈ M | bX = 0M} 6= 0M .

Hence the join of all elements a of L such that a 6= 0 and aX = 0M is contained in the join of

all associated prime elements of 0M , by Theorem 16.

ACKNOWLEDGEMENTS The authors thank the readers of European Journal of Pure and

Applied Mathematics, for making our journal successful.

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