Primary Decomposition in Lattice Modules

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

ISSN 1307-5543 – www.ejpam.com

Primary Decomposition in Lattice Modules

C S Manjarekar

_{, U N Kandale}

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 Lto 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} a b≤pimpliesa≤por b≤p. Ifa∈L,b∈L,(a:b)is the join of all elementscinLsuch that c b≤ a. A proper elementp ofL is said to be primary if a b≤ pimplies a≤ por bn≤ pfor some positive integern. Ifa∈L, the radical of a denoted bypa=_∨{x∈L|xn¶a,n∈Z+}. An element a ∈ L is called compact if a ¶ _∨

αbα implies a ¶ bα1∨bα2∨. . .∨bαn for some finite subset_{α₁,α₂, . . . ,α_n}. Throughout this paper, L denotes a multiplicative lattice which satisfies the ACC so that each element ofLis compact. Ifqis a primary element ofLthen

p_q=∨{x ∈L|xn¶q, for some integern}

is a prime element containing q. It is easily verified that, p_q is a minimal prime containing q [2]. The prime pq which is same aspq is called the prime associated with qand has the

properties, pk_q¶q¶p_qfor some integerkanda b¶qimpliesa¶qor b¶p_q.

An element a is said to have a primary decomposition if there exist primary elements q1,q2, . . . ,qn such thata=q1∧q2∧. . .∧qn. If someqi contains the meet of remaining ones

∗_{Corresponding author.}

Email addresses:smanjrekaryahoo.o.in(C. Manjarekar),ujwalabirajegmail.om(U. Kandale)

then thisq_ican be dropped from the primary decomposition. After deleting such primary com-ponents and combining the primary comcom-ponents 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. Leta=q₁∧q₂. . .∧q_nbe a reduced primary decomposition of a and let p₁,p₂, . . . ,p_n denotes the associated primes of q₁,q₂, . . . ,q_n re-spectively,which are also called associated primes ofa. A subsetC of{p1,p2, . . . ,pn}is called

isolated ifp_i∈C implies p_j∈C wheneverp_j¶p_i.

LetM be a complete lattice andLbe a multiplicative lattice. ThenM is called L-module or module over Lif there is a multiplication between elements of LandM written asaBwhere a∈LandB∈M which satisfies the following properties,

i) (_∨

αaα)A=∨αaαA∀aα∈L,A∈M ii) a(_∨

αAα) =∨αaAα∀a∈L,Aα∈M iii) (a b)A=a(bA)∀a,b∈L,A∈M iv) I B=B

v) 0B=0_M, for all a,aα,b∈LandA,Aα∈M,

where I is the supremum of L and 0 is the infimum of L. We denote by 0_M and I_M the least element and the greatest element ofM. The elements of Lwill generally be denoted by a,b,c, . . . and elements of M will generally be denoted byA,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 | x B ¶ A}. An L-module M is called a multiplication L-module if for every elementN∈M there exists an elementa∈Lsuch thatN=aIM [4].

A proper element N ofM is said to be prime if aX ¶N implies X ¶N or aI_M ¶ N that is a ¶(N : I_M)for every a ∈ L, X ∈ M. An element N < I_M in M is said to be primary if aX ¶ N impliesX ¶ N or anIM ¶ N that is an ¶(N : IM) for some integer n. An element

N of M is called a radical element if (N : I_M) = p(N:I_M). 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. ThenN is said to have a primary decomposition if there exist primary elementQ₁,Q₂, . . . ,Q_nsuch that

N = Q₁∧Q₂∧. . .Q_n. 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. IfQis primary then

p

Q=p(Q:I_M)is prime. We note that,p(N:I_M)may also be denoted bypN.

2. Primary Decomposition of Elements

Theorem 1. If Q is a primary element of a lattice module M thenp(Q:I_M)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 thenpa¶p.

Proof. Leta b¶p(Q:IM)and suppose b

p

(Q:IM). Then(a b)n=anbn¶(Q:IM)for

some positive integern. Now bp(Q:IM)implies bm(Q:IM)for any positive integerm.

In particularbn(Q:IM). AsQis primary,(an)k¶(Q:IM)for some positive integerk. That

isat¶(Q:IM)for some positive integert anda¶

p

(Q:IM). Therefore

p

(Q:IM)is prime.

Let a ¶ p. Take any x ¶ pa. Then xn ¶a ¶ p for some positive integern. As p is prime, x ¶pand hencepa¶p.

The following theorem gives the relation between meet of primary elements and their equal associated primes.

Theorem 2. If Q₁,Q₂, . . . ,Q_kare p-primary elements of a lattice module M then Q₁∧Q₂∧. . .∧Q_k is p-primary.

Proof. By hypothesisp(Q_i:I_M) =p, i=1, 2, . . . ,k. LetQ=Q_1∧Q_2∧. . ._∧Q_k. We have,

p

∧Q_i = p((∧Q_i):I_M) = p(Q₁:I_M)∧p(Q₂:I_M)∧. . .∧p(Q_k:I_M) = p. We show that ∧Qi is primary, where i = 1, 2, . . . ,k. Let aX ¶ Q = ∧Qi where a ∈ L, X ∈ M. Suppose,

X Q. Then X Qi for some i(1¶ i¶k). AsQi is primary, aX ¶Qi andX Qi implies

a¶p(Q_i:I_M) =p. That isa¶p(Q:I_M). 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:I_M) =_∧{p_|p is minimal prime containing(N:I_M)_}.

An elementN of a lattice moduleM is said to be meet irreducible if for any two elements A1andA2ofM,N=A1∧A2implies eitherA1=NorA2=N.

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

positive integer m such that A_n=A_m 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. Supposeτ_{is not empty. As M satifies ACC,}τ _{has a maximal element say N. AsN} ∈τ_{, N} is not irreducible. So there exists elements A₁ andA₂ of M such that N = A₁∧A₂ where N ₆=A₁, N ₆=A₂. So, N <A₁,N <A₂. This shows that bothA₁ andA₂ can be written as the meet of a finite number of meet irreducible elements ofM. So there are irreducible elements K₁,K₂, . . . ,K_mandK₁′,K₂′, . . . ,K_n′ ofM such thatA₁=K₁∧K₂∧. . .∧K_mand

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. Leta,b∈L, a b¶(Q:N)and suppose, a(Q:N). Asa(Q:N),aN Q. Also asa b¶(Q:N), a bN ¶Q. ButaN Q andQ is a primary element implies that bn ¶ (Q : IM) for some integer n. But bnIM ¶Q

implies bnN ¶Q. Hence, b¶p(Q:N). Therefore,(Q:N)is a primary element of L. Now sinceNQ, there existsA∈M andA¶N such thatAQ. Leta¶p(Q:N). ThenanN¶Q. Hence, anA¶Q. ButAQ andQ is primary implies that(an)k = am ¶(Q : I_M) for some integerm. That isa¶p(Q:I_M) =pandp(Q:N)¶p. Conversely, leta¶p(Q:I_M) =p. Hence,anIM ¶Qfor some integern. SoanN¶Q for some integern. Thusan¶(Q:N)and

a¶p(Q:N). This shows thatp¶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 ap then(Q:a) =Q.

Proof. Supposea p where p =p(Q:I_M). Since, a pthere is some b¶ asuch that b p. Let X ¶(Q :a). Then aX ¶Q and hence bX ¶Q where bp(Q:I_M) = p. AsQ is a p-primary, X ¶Q. Hence, (Q:a)¶Q. Conversely letX ¶Q. Sincea ¶1,aX ¶Q. So X ¶(Q:a)and henceQ¶(Q:a). Therefore,Q= (Q:a).

The following theorem gives the characterisation of a prime elementpofLto be equal to some associated prime of an element which has a primary decomposition.

Theorem 7. Let N6=I_M 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=pQ_i for some i if and only if(N :X)is a p-primary element of L for some X N .

Proof. LetN=Q1∧Q2∧. . .∧Qkbe a reduced primary decomposition ofN. First suppose

that, p = pQ_i for some i. Without loss of generality we can assume that p = p(Q₁:I_M)

where pi =

p

(Qi:IM)i=1, 2, . . . ,k. We prove that,(N:X)is a p-primary element of Lfor

someX N. Since the decomposition is reducedQ_iQ₁∧Q₂∧. . .∧Q_i−1∧Q_i+1∧. . .∧Qk

fori=1, 2, . . . ,k. In particular,Q₁ Q₂∧Q₃∧. . .∧Q_k. So there existsX ¶Q₂∧Q₃∧. . .∧Q_k such thatX Q1and henceX N=Q1∧Q2∧. . .∧Qk. Also

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

i=2, 3, . . . ,k. So 1¶(Q_i:X). But(Q_i:X)¶1 implies(Q_i:X) =1 fori=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 ofLwhereX N. Conversely assume that(N:X)is a p-primary element of Lfor someX N,X ∈M. We prove thatpQ_i=pfor somei. We have, p=p(N:X) =p[(Q1∧Q2∧. . .∧Qk):X]=

p

(Q1:X)∧

p

(Q2:X)∧. . .∧

p

(Qk:X). We

claim that for eachi,p(Qi :X) =pi or 1 and equal topi for at least onei. We have

X N =Q1∧Q2∧. . .∧QkimpliesX Qi for at least onei(1¶i¶k). Suppose

X Q_r(1¶r¶k)andX ¶Q₁∧Q₂∧. . .∧Q_r−1∧Q_r+1∧. . .Qkthat isX ¶∧Qi, where(i6=r).

We have,aX ¶Q_i for alli6= r anda∈ L. Hence,a¶p(Q_i:X) for alla∈ L. In particular, 1¶p(Q_i:X)for all i6= r. But,p(Q_i:X) ¶1 for all i6= r. Therefore,p(Q_i:X) =1 for alli6= r. Fori= r,X Qr. Let a¶

p

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

n, where X Qr. AsQr is primary, an ¶

p

(Qr:IM) = pr. Thus, a¶ pr, since pr is prime

and we have, p(Q_r:X)¶p_r. On the other hand, let a¶ p_r =pQ_r=p(Q_r:I_M). Hence, an ¶(Qr :IM) for some positive integern. That is anIM ¶Qr and therefore, anX ¶Qr, for

some positive integern. Consequently, an ¶ (Q_r : X)and hence a¶ p(Q_r:X). This gives p_r ¶ p(Q_r:X). Hence, p(Q_r:X) = p_r where X Q_r. We have shown that for each i,

p

(Qi:X) =pi or 1 and is equal topi for at least onei, sinceX N. Then,

p=p(N :X) =p(Q₁:X)_∧. . .∧p(Q_k:X)

is the meet of some of the prime elementsp1,p2, . . . ,pl (1¶l ¶k). That is

p=p(N:X) =p₁∧p₂∧. . .∧p_l.

We show that p = p_i for some i. We have, p ¶ p_i i= 1, 2, . . . ,l. If for eachi, p 6= 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,x1x2. . .xl ¶p1∧p2∧. . .∧pl =p. This shows that xi ¶pfor at least one

i(1¶i¶k)a contradiction. Hence,p=p_i for at least onei. This leads us to the following result.

Theorem 8. Let N6=I_M be an element of a lattice module M and assume that N has a primary decomposition. If N =Q₁∧Q₂∧. . .∧Q_m=S₁∧S₂∧. . .∧S_n are two reduced primary decom-positions of N then n=m and the Qi and Si can be so numbered that

p

(Qi: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 Land associated primes of zero.

Proof. Let 0=q1∧q2∧. . .∧qk be a reduced primary decomposition of 0 andpi =pqi,

i = 1, 2, . . . ,k. Suppose a is a zero divisor. Then if a = 0 obviously a ¶ p₁∨p₂∨. . .∨p_k. Suppose,ais a proper zero divisor that isa₆=0 and leta b=0 where b₆=0. Now,

a b=0¶q1∧q2∧. . .∧qk={0}. Hence,a b¶qifor alliandbqifor at least onei. Because,

b ¶q_i for all i implies b¶ q₁∧ q₂∧. . .∧q_k = {0} and hence b= 0 , a contradiction. Let bq_j. Then,a b¶q_j,bq_j andq_j is a primary element. Therefore, a¶p

q_j= p_j, which shows thata¶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∧. . .∧Q_m. If every Q_i (1¶i¶m)is a prime element then

(N:IM) =

p

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

Proof. Suppose eachQ_i is a prime element. Leta¶p(N:I_M). Then anIM ¶N=Q1∧Q2∧. . .∧Qm

for some positive integern. This implies that,anI_M ¶Q_i for eachi. AsQ_i is a prime element, aIM ¶Qi or an−1IM ¶Qi. If aIM ¶Qi then a¶(Qi : IM). Otherwise an−1IM ¶Qi implies

aI_M ¶ Q_i or an−2I_M ¶ Q_i. Continuing in this way we obtain, a ¶ (Q_i : I_M) for each i. Thereforea¶(Q₁:I_M)∧(Q₂:I_M)∧. . .∧(Q_m:I_M). That isa¶(N:I_M)and hence

(N:IM) =

p

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

p

(N:IM). We show that

(Qi:IM) =pi. Let y¶pi =

p

(Qi:IM). As m

∧

i=1pi is irredundant(reduced) there existsz¶∧pj

such thatz p_i in L. Now yz¶ _∧m

i=1pi =

m

∧

i=1

p

(Q_i:I_M) = ∧m

i=1(Qi :IM)implies yzIM ¶Qi for

eachi. SinceQ_i is primary,z p_i gives y ¶(Q_i :I_M). Hence, p_i ¶(Q_i :I_M). Consequently, p_i= (Q_i:I_M).

Now we obtain a characterization of a prime element p of Lcontaining some associated primep_i ofN6=I_M in a lattice moduleM.

Theorem 11. Let N 6=I_M have a reduced primary decomposition N =Q₁∧Q₂∧. . .∧Q_m and pi=

p

(Qi :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 p_i for some i.

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

i=1(Qi : IM) implies (N : IM)¶ p.

Conversely assume that(N:I_M)¶p. Then(Q₁:I_M)∧(Q₂:I_M)∧. . .∧(Q_m:I_M)¶pimplies

(Qi : IM) ¶ p for some i. But

p

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

Hence,p_i¶pfor somei.

In our next result we show that thoseQ′_is can be uniquely determined which are isolated primary components ofN6=IM.

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

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 elementA∈ {X ∈M |(N :X)p_i}. Then(N :A) p_i. So there exists a∈ Lsuch thataA¶N anda pi =

p

(Qi:IM). Hence,anIM Qi for any integern. Now

aA¶Q_i,anI_MQ_iandQ_iis primary givesA¶Q_i. HenceQ_i′¶Q_iand the first part is proved. If p_i is a minimal associated primes of N it follows that p_j p_i fori₆= j. Then there exists b_j¶p_jinLsuch thatb_jp_i. We haveb_j¶p_j =p(Q_j:I_M) =∨{a_j∈L|as_jjI_M ¶Q_jfor some integer s_j}. Since each element of L is compact, we have b_j ¶ p_j = ∨n

j=1{aj | a

sj

j IM ¶Qj for

some integersj}. Puts1+s2+. . .+sn=k(j). Thenbjk(j)IM ¶(a1∨a2∨. . .∨an)k(j)IM¶Qj.

Clearly b= Π

j6=ibj

k(j)_p

i aspi is prime. However, bIM ¶ ∧

j6=iQj. Next take anyX ¶Qi. Then

X bI_M¶ m_∧

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

X ¶_∨{X ∈M |(N:X)pi}=Qi

′

and we haveQi¶Qi

′

. Consequently,Qi=Qi

′

.

We now relate the radical of N with the isolated primes ofN ∈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∧. . .∧Q_n thenp(N:I_M)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∧. . .∧p_n,

where pi=

p

(Qi:IM)are associated primes ofN. If some pk is not isolated thenpk≥pi for

somep_i and hence we can delete such elements from the above primary decomposition and we are through.

We note that an element A=aI_M ofM where a∈ Lis said to be nilpotent ifanI_M =0_M for some positive integern. If a lattice module M satisfies the ACC and if every element of L is the join of meet prncipal elements then every element ofM 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 aI_M is nilpotent is the meet of the isolated primes of0_M.

Proof. Let 0_M= ∧n

i=1Qi be a reduced primary decomposition of 0M andpi=

p

(Q_i:I_M)be an associated prime ofQ_i,i=1, 2, . . .n. We have

p

andp(0_M:I_M) =p₁∧p₂∧. . .∧p_kwhere p₁,p₂, . . . ,p_kare the isolated primes of 0_M. Hence,

p

(0_M:I_M)is the meet of isolated primes of 0_M.

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

Theorem 15. For A∈M ,p(A:I_M)is prime if and only if A has a single isolated prime element. Proof. Let A= Q₁∧Q₂∧. . .∧Q_n be primary decomposition ofA. If Ahas a single iso-lated prime element p thenp(A:IM) = p. Conversely, assume that

p

(A:IM) is prime and

p

(A:I_M) =p₁∧p₂ where p₁,p₂ are isolated primes ofA. Then there are x,y ∈L such that x ¶p₁, x p₂ and y ¶p₂, y p₁. Hence x y ¶p(A:I_M)which is prime. But then x ¶p₂ or y¶p1 which is a contradiction. ThusAhas a single isolated prime element.

In the remaining part we assume that a lattice module M satisfies the ACC over a multi-plicative latticeLin which every element is the join of meet principal elements. This condition assures that any elementM has a reduced primary decomposition.

Theorem 16. Let A be any element of M and b∈L be such that A6=I_M. Then A= (A:b)if and only if b is contained in no associated prime element of A.

Proof. Let A=Q₁∧Q₂∧. . .∧Q_m be an irredundant primary decomposition of Aand let pi =

p

(Qi:IM). Supposebpi for anyi=1, 2, . . . ,m. This leads us to the fact bnpi for

any positive integern. We know that(A: b)b¶A[3]and thus(A: b)b¶Q_i for alli. Since Qi is primary and bpi we have(A: b)¶Qi. That is (A:b)¶

m

∧

i=1Qi =A. ButA¶(A: b)

gives (A: b) = A. Conversely, suppose(A: b) =Aand if possible without loss of generality assume that b¶ p₁. Then(Q: bs) =I_M for some integers. We have(A:b): b=A: b2 [3]. Continuing in this wayA:b=A:bs. ButA:b=AimpliesA:bs=A. Finally,

A=(A:bs) = ((Q₁∧Q₂∧. . .∧Q_m):bs) = ((Q₁:bs)∧(Q₂:bs)∧. . .∧(Q_m:bs) = _∧

j6=1(Qj:b

s_{)≥ ∧}

j6=1Qj≥A.

That isA= _∧

j6=1Qj. This contradicts the fact thatA=

m

∧

i=1Qi is a reduced primary decomposition

ofA.

The above theorem can be restated in the following form.

Theorem 17. Let N 6= I_M have a reduced primary decomposition Q₁∧Q₂ ∧. . .∧Q_m and p₁,p₂, . . . ,p_m be the associated primes of Q′_is. 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.

An elementX ∈M is called a zero divisor if(0_M:X)6=0 so there existsa6=0 in L such thataX =0_M.

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 a₆=0and aX =0_M is contained in the join of all associated prime elements of0M.

Proof. LetX be a zero divisor of M. Then 0_M :X ₆=0 that is there existsa₆=0 in L such thataX =0_M. We know that for an elementbofL(b6=0,bX =0_M)to be contained in some associated prime of 0_M it is necessary and sufficient that

(0_M:b) =_∨{X ∈M |bX=0_M} 6=0_M.

Hence the join of all elementsaof Lsuch thata6=0 andaX =0_M is contained in the join of all associated prime elements of 0_M, by Theorem 16.

ACKNOWLEDGEMENTS The authors thank the readers of European Journal of Pure and Applied Mathematics, for making our journal successful.

References

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