Article
Kyungpook Mathematical Journal 2021; 61(1): 3348
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
OnWeakly Prime andWeakly 2absorbing Modules over Noncommutative Rings
Nico J. Groenewald
Department of Mathematics, Nelson Mandela University, Port Elizabeth, South Africa
email : nico.groenewald@mandela.ac.za
Received: March 22, 2020; Revised: September 8, 2020; Accepted: October 8, 2020
Abstract
Most of the research on weakly prime and weakly 2absorbing modules is for modules over commutative rings. Only scatterd results about these notions with regard to noncommutative rings are available. The motivation of this paper is to show that many results for the commutative case also hold in the noncommutative case. Let
Keywords: 2absorbing submodule, weakly 2absorbing submodule, prime submodule, weakly prime submodule, weakly prime radical.
1. Introduction
In 2007 Badawi [3] introduced the concept of 2absorbing ideals of commutative rings with identity, which is a generalization of prime ideals, and investigated some properties. He defined a 2absorbing ideal
Throughout this paper, all rings are associative with identity elements (not necessarily commutative) and modules are unitary left modules. Let
Following [9] a proper ideal
From [10] A proper ideal
Definition 1.1.
([1, Definition 3.3]) Let
Remark 1.2.
Let
Compare the next Theorem with [2, Corollary 2.3].
Theorem 1.3.
Let
Compare (1)
Theorem 1.4.
Let

(1)
N is a weakly prime submodule ofM . 
(2) For a left ideal
P ofR and submoduleD ofM with$0\ne PD\subseteq N$ , either$P\subseteq (N{:}_{R}M)$ or$D\subseteq N.$ 
(3) For any element
$a\in R$ and$L\le M$ , if$0\ne aRL\subseteq N$ , then$L\subseteq N$ or$a\in (N{:}_{R}M)$ . 
(4) For any right ideal
I ofR and$L\le M$ , if$0\ne IL\subseteq N$ , then$L\subseteq N$ or$I\subseteq (N{:}_{R}M).$ 
(5)] For any element
$a\in R$ and$L\le M$ , if$0\ne RaRL\subseteq N$ , then$L\subseteq N$ or$a\in (N{:}_{R}M).$ 
(6)] For any element
$a\in R$ and$L\le M$ , if$0\ne RaL\subseteq N$ then$L\subseteq N$ or$a\in (N{:}_{R}M)$ .
(1)
(2)
(2)
(3)
(3)
Remark 1.5.
From [5] we know that if
Proposition 1.6.
Let
From [12] we have that
Proposition 1.7.
Let
Remark 1.8.
The converse of Proposition 1.7 is not true in general. Suppose that
Lemma 1.9.
Let
The following result gives characterizations of weakly prime submodules.
Theorem 1.10.
Let

(1)
P is a weakly prime submodule ofM . 
(2)
(P:Rx)=(P:M) ∪ (0:Rx) for anyx ∈ MP . 
(3)
(P:Rx)=(P:M) or(P:Rx)=(0:Rx) for anyx ∈ MP .
Proposition 1.11.
Let

(1)
N=Q ⊕ M_{2} is a weakly prime submodule ofM if and only ifQ is a weakly prime submodule ofM_{1} andr ∈ R ,x ∈ M_{1} withrRx=0 , butx ∉ Q ,r∉(Q:M_{1}) impliesrM_{2}=0. 
(2)
N=M_{1} ⊕ Q is a weakly prime submodule ofM if and only ifQ is a weakly prime submodule ofM_{2} andr ∈ R ,x∈ M_{2} withrRx=0 , butx ∉ Q ,r∉(Q:M_{2}) impliesrM_{1}=0.
Remark 1.12.
Let
Proposition 1.13.
Let
Remark 1.14.
The converse of Proposition 1.13 is not true in general as the following example shows.
Example 1.15.
Suppose
2. The Weakly Prime Radical
We begin this section with the definition of weakly msystems.
Definition 2.1
Let
Proposition 2.2.
Let
The following proposition offers several characterizations of a weakly msystem
Proposition 2.3.
Let

(1)
P is weakly prime; 
(2)
S is a weakly msystem; 
(3) for each left ideal
$A\subseteq R$ , and for every submodule$L\leqq M$ , if$L\cap S\ne \varnothing $ ,$AM\cap S\ne \varnothing $ and$AL\ne 0$ then$AL\cap S\ne \varnothing $ ; 
(4) for each ideal
$A\subseteq R$ , and for every$m\in M$ , if$Rm\cap S\ne \varnothing $ ,$AM\cap S=\varnothing $ and$AL\ne 0$ , then$ARm\cap S\ne \varnothing $ ; 
(5) for each
a∈ R , and for each$m\in M$ , if$Rm\cap S\ne \varnothing $ ,$aM\cap S\ne \varnothing $ and$aRm\ne 0$ , then$aRm\cap S\ne \varnothing .$
Proposition 2.4.
Let
Next we need a generalization of the notion of
Definition 2.5.
Let
Theorem 2.6.
Let
3. Weakly 2absorbing Submodules
From [11] we have the following:
Definition 3.1.
Let
Definition 3.2.
Let
Remark 3.3.
If
We now have the following:
Definition 3.4.
Let
Remark 3.5.
Every 2absorbing submodule is weakly 2absorbing but the converse does not necessarily hold. For example consider the case where
Proposition 3.6.
Let
Proposition 3.7.
Let
Compare the following theorem with that of [7, Theorem 2.3(ii)].
Theorem 3.8.
The intersection of each pair of weakly prime submodules of an

(1)
am ∉ N andab ∉ (N:_{R}M) ; 
(2)
am ∉ N andab ∉ (K:_{R}M) ; 
(3)
am ∉ K andab ∉ (N:_{R}M) ; 
(4)
am ∉ K andab ∉ (K:_{R}M) .
We first consider Case(1). Since
Definition 3.9.
Let
The following result is an analogue of [6, Theorem 1].
Theorem 3.10.
Let

(1)
aRbN=a(N:_{R}M)m=b(N:_{R}M)m=0. 
(2)
a(N:_{R}M)N=b(N:_{R}M)N=(N:_{R}M)bN=(N:_{R}M)bm=(N:_{R}M)^{2}m=0.
(1) Assume that
(2) Assume that
Consequently,
The following result is an analogue of [6, Lemma 1].
Proposition 3.11.
Assume that N is a weakly 2absorbing submodule of an
Thus
4. On a Question from Badawi and Yousefian
In [4], the authors asked the following question:
This section is devoted to studying the above question and its generalization in modules over noncommutative rings.
Definition 4.1.
Let
The following result and its proof are analogous of [6, Lemma 2].
Lemma 4.2.
Let
Let
The following result and its proof are analogous of [6, Theorem 1] and its proof.
Theorem 4.3.
Assume that
5. Weakly 2absorbing Submodules of Product Modules
Proposition 5.1.
Let

(1)
N_{1} is a 2absorbing submodule ofM_{1} ; 
(2)
N_{1} × M_{2} is a 2absorbing submodule ofM_{1} × M_{2}; 
(3)
N_{1} × M_{2} is a weakly 2absorbing submodule ofM_{1} × M_{2}.
Proposition 5.2.
Let

(1)
N_{2} is a 2absorbing submodule ofM_{1} ; 
(2)
M_{1} × N_{2} is a 2absorbing submodule ofM_{1} × M_{2}; 
(3)
M_{1} × N_{2} is a weakly 2absorbing submodule ofM_{1} × M_{2} .
Proposition 5.3.
Let
Proposition 5.4.
Let
Proposition 5.5.
Let
If

(1)
N_{1} is a weakly 2absorbing submodule ofM_{1} , 
(2)
N_{2} is a weakly 2absorbing submodule ofM_{2} .

(1) Suppose that
${N}_{1}\times {N}_{2}$ is a weakly 2absorbing submodule of${M}_{1}\times {M}_{2}$ . Let${a}_{1},{a}_{2}\in {R}_{1}$ and$m\in {M}_{1}$ such that$0\ne {a}_{1}{R}_{1}{a}_{2}{R}_{1}m\subseteq {N}_{1}$ . Clearly,$(0,0)\ne ({a}_{1},1)({R}_{1}\times {R}_{2})({a}_{2},1)({R}_{1}\times {R}_{2})(m,{m}_{2})$ for any${m}_{2}\in {N}_{2}.$ Hence$(0,0)\ne ({a}_{1},1)({R}_{1}\times {R}_{2})({a}_{2},1)({R}_{1}\times {R}_{2})(m,{m}_{2})\subseteq {a}_{1}{R}_{1}{a}_{2}{R}_{1}m\times 1{R}_{2}1{R}_{2}{m}_{2}\subseteq {N}_{1}\times {N}_{2}.$ Since${N}_{1}\times {N}_{2}$ is a weakly 2absorbing submodule of${M}_{1}\times {M}_{2}$ ,$({a}_{1},1)({a}_{2},1)\in ({N}_{1}\times {N}_{2}:{M}_{1}\times {M}_{2})$ or$({a}_{1},1)(m,{m}_{2})\in {N}_{1}\times {N}_{2}$ or$({a}_{2},1)(m,{m}_{2})\in {N}_{1}\times {N}_{2}.$ Consequently${a}_{1}{a}_{2}\in ({N}_{1}:{M}_{1})$ or${a}_{1}m\in {N}_{1}$ or${a}_{2}m\in {N}_{1}$ . HenceN_{1} is a weakly 2absorbing submodule ofM_{1} . 
(2) This follows as in part (1).
The converse of the above proposition is no true in general:
Example 5.6.
Suppose that
Proposition 5.7.
Let

(1)
N_{1} △ M_{2} is a weakly 2absorbing submodule ofM_{1} △ M_{2}; 
(2)

(a)
N_{1} is a weakly 2absorbing submodule ofM_{1}; 
(b) For each
a_{1},a_{2} ∈ R andm ∈ M_{1} such thata_{1}Ra_{2}Rm=0 ifa_{1}a_{2}∉(N_{1}:M_{1}) anda_{1}m ∉ N_{1} anda_{2}m ∉ N_{1} thena_{1}Ra_{2}M_{2}=0.

(a) Suppose
(b) Let
(2)
Let
Proposition 5.8.
Let
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