# Notes on the Union of Weakly Primary Submodules

Authors: Peyman Ghiasvand; Farkhonde Farzalipour
Journal: ISRN Discrete Mathematics
 (2011)

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## Abstract

LetR be a commutative ring with identity, and let M be an R-module. A proper submodule N of M is said to be weakly primary if 0≠rm∈N for r∈R and m∈M, which implies that either m∈N or rnM⊆N for some positive integer n. In this paper, we study weakly primary submodules, and we investigate the union of weakly primary submodules of R-modules.


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## Body

## 1. Introduction

LetR be a commutative ring with identity, and let M be a unital R-module. A commutative ring R is called a quasilocal ring if it has a unique maximal ideal P and denoted by (R,P). Let N be a submodule of M and the ideal {r∈R:rM⊆N} denoted by (N:M). Let I be an ideal of R and the radical of I denoted by I and defined I={r∈R:rn∈Iforsomepositiveintegern}. A proper submodule N of M is said to be prime (weakly prime) if rm∈N (0≠rm∈N), then either m∈N or rM⊆N (either m∈N or rM⊆N), where r∈R and m∈M. A proper submodule N of M is said to be primary (weakly primary) if rm∈N (0≠rm∈N), then either m∈N or rnM⊆N for some positive integer n (either m∈N or rnM⊆N for some positive integer n), where r∈R and m∈M. It is clear that every primary submodule is weakly primary. However, since 0 is always weakly primary (by definition), so a weakly primary submodule need not be primary. A proper submodule N of an R-module M said to be maximal if there is no submodule K of M such that N⊈K⊈M. A submodule N of M is called u-submodule of M, provided that N contained in a finite union of submodules must be contained one of those submodules. M is called u-module if every submodule of M is a u-submodule. A submodule N of M is called upr-submodule of M, provided that N contained in a finite union of primary submodules that must be contained in one of those primary submodules. M is called upr-module if every submodule of M is a upr-submodule. A submodule N of M is called um-submodule of M, provided that N contained in a finite union of maximal submodules that must be contained in one of those submodules.M is called um-module if every maximal submodule of M is a um-submodule. An R-module M is called a multiplication module, provided that for each submodule N of M, there exists an ideal I of R such that N=IM. If R is a ring and M an R-module, the subset T(M) of M is defined by T(M)={m∈M:rm=0forsome0≠r∈R}. Obviously, if R is an integral domain, then T(M) is a submodule of M. In this paper, we investigate finite unions of weakly primary submodules of R-modules.


## 2. On Weakly Primary Submodules

It is clear that every primary submodule is a weakly primary submodule. However, since 0 is always weakly primary (by definition), a weakly primary submodule need not be primary, but we have the following results.Proposition 2.1.
LetM be an R-module with T(M)=0. Then, every weakly primary submodule of M is primary.Proof.
LetN be a weakly primary submodule of M. Suppose that rm∈N, where r∈R, m∈M. If 0≠rm∈N, N weakly primary gives m∈N or rnM⊆N for some positive integer n. If rm=0, then r=0 or m=0, since T(M)=0. So, N is primary.Proposition 2.2.
LetM be a module over a quasilocal ring (R,P) with PM=0. Then, every proper submodule of M is weakly primary.Proof.
LetN be a proper submodule of M and 0≠rm∈N, where r∈R and m∈M. If r is a unite, then m∈N. Let r is not a unite, so rm∈PM=0, a contradiction. Hence, N is weakly primary.Lemma 2.3.
LetM be an R-module. Assume that N and K are submodules of M such that K⊆N with N≠M. Then, the following hold:(i)
ifN is a weakly primary submodule of M, then N/K is a weakly primary submodule of M/K,(ii)
ifK and N/K are weakly primary submodules, then N is weakly primary.Proof.
(i) Let0≠r(m+K)=rm+K∈N/K, where r∈R and m∈M. If rm=0, then r(m+K)=0, which is a contradiction. If rm≠0, N weakly primary gives either m∈N or rn∈(N:RM) for some positive integer n, hence either m+K∈N/K or rn∈(N/K:RM/K) (since we have (N:RM)=(N/K:RM/K)), as required.
(ii) Let0≠rm∈N, where r∈R and m∈M, so r(m+K)=rm+K∈N/K. If rm∈K, then K weakly primary gives either m∈K⊆N or rn∈(K:RM)⊆(N:RM). So, we may assume that rm∉K. Then, 0≠r(m+K)∈N/K. Since N/K is weakly primary, we get either m+N∈N/K or rn∈(N/K:RM/K)=(N:M) for some positive integer n. Thus, m∈N or rn∈(N:RM) for some positive integer n, as required.Theorem 2.4.
LetM be a secondary R-module and N a nonzero weakly primary R-submodule of M. Then, N is secondary.Proof.
Letr∈R. If rnM=0 for some n∈N. Then, rnN⊆rnM=0, so r is nilpotent on N. Suppose that rM=M; we show that r divides N. Assume that n∈N. So, n=rm for some m∈M. We may assume that 0≠rm. Hence, 0≠rm∈N and rnM⊈N for any positive integer n (since rM=M), then N weakly primary gives m∈N. Thus, rN=N, as needed.Theorem 2.5.
LetM be an R-module, N a secondary R-submodule of M, and K a weakly primary submodule of M. Then, N∩K is secondary.Proof.
The proof is straightforward.Proposition 2.6.
LetM be a module over a commutative ring R and S a multiplicatively closed subset of R. Let N be a weakly primary submodule of M such that (N:M)∩S=∅. Then, S-1N is a weakly primary submodule of S-1R-module S-1M.Proof.
Let0/1≠r/s·m/t∈S-1N, where r/s∈S-1R and m/t∈S-1M. So, 0/1≠rm/st=n/t′ for some n∈N and t′∈S, hence there exists s′∈S such that 0≠s′t′rm=s′stn∈N (because if s′t′rm=0, rm/st=s′t′rm/s′t′st=0/1, a contradiction) and s′t′∉(N:M), so N weakly primary gives 0≠rm∈N. Hence, m∈N or rn∈(N:M), thus (r/s)n∈S-1(N:RM)⊆(S-1N:S-1RS-1M) or m/t∈S-1N, as needed.Lemma 2.7.
LetM be a module over a quasilocal ring (R,P) and N a weakly primary submodule of M. Then, (N:RM)P=(NP:RPMP).Proof.
Letr/s∈(NP:RPMP) and m∈M. We show that rm∈N. We may assume that rm≠0. We have r/s·m/1∈S-1N, so rm/s=n/t for some t∈S and n∈N. There exists t′∈S such that t′trm=t′sn∈N. If t′trm=0, then tt′∈(0:rm)∩S⊆P∩S=∅, a contradiction. So, 0≠tt′rm∈N and tt′∉(N:M), then rm∈N. Thus, (NP:RPMP)⊆(N:RM)P. Clearly, (N:RM)P⊆(NP:RPMP), so the proof is complete.Theorem 2.8.
LetM be a module over a quasilocal ring (R,P). Then, there exists a one to one correspondence between the weakly primary submodules of M and the weakly primary submodules of RP-module MP.Proof.
LetK be a weakly primary submodule of MP. So, K=NP for some submodule N of M. We show that N is weakly primary submodule of M. Let 0≠rm∈N, so 0/1≠rm/1∈NP (if rm/1=0/1, then srm=0 for some s∈S, s∈(0:rm)∩S⊆P∩S=∅, a contradiction). Hence, (r/1)n∈(NP:RPMP)⊆(N:PM)P for some positive integer n by Lemma 2.7 or m/1∈NP, since NP is weakly primary. Thus, rn∈(N:M) for some positive integer n or m∈N, as required. Let N be a weakly primary submodule of M. Then, by Proposition 2.6, NP is weakly primary submodule of MP.


## 3. Unions of Weakly Primary Submodules

Definition 3.1.
LetM be a module over a commutative ring R and N a submodule of M; N is called a uwpr-submodule of M, provided that N contained in a finite weakly primary submodules of M must be contained in one of those weakly primary submodules. M is called a uwpr-module if every submodule of M is a uwpr-submodule.Clearly, everyu-module is uwpr-module, and every uwpr-module is upr-module and also, every upr-module is um-module.Theorem 3.2.
LetM be a module over a commutative ring R and K a submodule of M. Then,(i)
ifN is a uwpr-submodule of M and K is a weakly primary submodule of M such that K⊆N, then N/K is a uwpr-submodule of R-module M/K,(ii)
ifN/K is a uwpr-submodule of M/K, then N is a uwpr-submodule of M.Proof.
(i) LetN/K⊆L1∪L2∪⋯∪Ln, where Li’s are weakly primary submodules of M/K. Then, by Lemma 2.3, there exists weakly primary submodules Ni of M such that Li=Ni/K for i=1,2,…,n. So, N/K⊆N1/K∪⋯∪Nn/K=N1∪⋯∪Nn/K, so N⊆N1∪N2∪⋯∪Nn. Hence, N⊆Nk for some k, since N is uwpr-submodule. Thus N/K⊆Nk/K, for some k, as needed.
(ii) LetN⊆N1∪N2∪⋯∪Nn, where Ni’s are weakly primary submodule of M. So, N/K⊆N1∪⋯∪Nn/K=N1/K∪⋯∪Nn/K. Therefore, Ni/K’s are weakly primary submodules of M/K, so N/K⊆Nk/K for some k. Hence, N⊆Nk for some k, as required.Theorem 3.3.
LetM be an R-module, N a weakly primary submodule of M, and M a uwpr-module. Then, M/N is a uwpr-module.Proof.
By Theorem3.2.Theorem 3.4.
LetM be a module over quasi local ring (R,P). Then, M is a uwpr-module if and only if RP module MP is a uwpr-module.Proof.
LetM be a uwpr-module, and let K be a submodule of RP-module MP such that K⊆K1∪⋯∪Kn, where Ki’s are weakly primary submodules of MP. So, K=NP for some submodule N of M and Ki=(Ni)P for some weakly primary submodules Ni of M by Theorem 2.8. Hence, NP⊆(N1)P∪⋯∪(Nn)P, so NP⊆(N1∪⋯∪Nn)P, thus N⊆N1∪⋯∪Nn by [1, Theorem  2.8]. Therefore, by hypothesis N⊆Nk for some 1≤k≤n. So NP⊆(Nk)P for some 1≤k≤n as needed.
Conversely, letRP-module MP is a uwpr-module, and let N be a submodule of M such that N⊆N1∪⋯∪Nn, where Ni’s are weakly primary submodules of M. So, NP⊆(N1∪⋯∪Nn)P=(N1)P∪⋯∪(Nn)P. Thus, NP⊆(Nk)P for some 1≤k≤n by hypothesis. Then, N⊆Nk for some 1≤k≤n. So M is a uwpr-module.Theorem 3.5.
LetM be a finitely generated R-module. Then, M is a um-module if and only if every submodule N in M such that N⊆⋃i=1nPi, where Pi’s are weakly primary submodules implies that N+Pi≠M for some i.Proof.
LetM be a finitely generated um-module. Suppose that N be a submodule of M such that N⊆⋃i=1nPi, where Pi’s are weakly primary submodules of M. For each Pi, Mi is a maximal submodule containing Pi. Then, N⊆⋃i=1nMi, and so, N⊆Mi for some i by hypothesis. Since Pi⊆Mi, we have N+Pi⊆Mi≠M.
Conversely, letN be a submodule of M such that N⊆⋃i=1nMi, where Mi’s are maximal submodules of M. Since every maximal submodule is weakly primary submodule, then N+Mi≠M for some i by hypothesis. Therefore, since Mi⊆N+Mi⊈M, then N+Mi=Mi, so N⊆Mi for some i. The proof is complete.Definition 3.6.
By a chain of weakly primary submodules of anR-module M, we mean a finite strictly increasing sequence P1⊆⋯⊆Pn; the weakly primary dimension of this chain is n. We define the weakly primary dimension of M to be the supremum of the lengths of all chains of weakly primary submodules in M.Theorem 3.7.
LetM be a finitely generated R-module with weakly primary dimension 1. Then, M is a uwpr-module if and only if M is a um-module.Proof.
LetM be a uwpr-module. Since every uwpr-module is a um-module, so M is a um-module.
Conversely, letM be a um-module. Let N be a nonzero submodule of M such that N⊆P1⋃P2⋃⋯⋃Pn, where Pi’s are weakly primary submodules of M. We may assume that Pi≠0 for all i∈{1,2,…,n}. By Theorem 3.5, N+Pi≠M for some i. There exists a maximal submodule Pi of M such that N+Pi⊆Mi. Since 0 is a weakly primary submodule of M, so we have 0⊈Ni⊆Mi. Hence, since weakly primary dimension of M is 1; Pi=Mi. Consequently, N⊆Pi for some i, as needed.Theorem 3.8.
LetM be an R-module with T(M)=0. Then, M is a upr-module if and only if M is a uwpr-module.Proof.
LetM be a uwpr-module. Then, M is a upr-module since every primary submodule is weakly primary. Let M be a up-module and N a submodule of M such that N⊆P1⋃P2⋃⋯⋃Pn where Pi’s are weakly primary submodules of M. By Proposition 2.1, Pi’s  are primary submodule, so N⊆Pi for some i, as needed.



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*Source: 939687-2011-12-28.xml*