Article
Kyungpook Mathematical Journal 2021; 61(1): 33-48
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
OnWeakly Prime andWeakly 2-absorbing Modules over Non-commutative Rings
Nico J. Groenewald
Department of Mathematics, Nelson Mandela University, Port Elizabeth, South Africa
e-mail : 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 2-absorbing modules is for modules over commutative rings. Only scatterd results about these notions with regard to non-commutative rings are available. The motivation of this paper is to show that many results for the commutative case also hold in the non-commutative case. Let
Keywords: 2-absorbing submodule, weakly 2-absorbing submodule, prime submodule, weakly prime submodule, weakly prime radical.
1. Introduction
In 2007 Badawi [3] introduced the concept of 2-absorbing ideals of commutative rings with identity, which is a generalization of prime ideals, and investigated some properties. He defined a 2-absorbing 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, either or -
(3) For any element
and , if , then or . -
(4) For any right ideal
I ofR and, if , then or -
(5)] For any element
and , if , then or -
(6)] For any element
and , if then or .
(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 ∈ M-P . -
(3)
(P:Rx)=(P:M) or(P:Rx)=(0:Rx) for anyx ∈ M-P .
Proposition 1.11.
Let
-
(1)
N=Q ⊕ M2 is a weakly prime submodule ofM if and only ifQ is a weakly prime submodule ofM1 andr ∈ R ,x ∈ M1 withrRx=0 , butx ∉ Q ,r∉(Q:M1) impliesrM2=0. -
(2)
N=M1 ⊕ Q is a weakly prime submodule ofM if and only ifQ is a weakly prime submodule ofM2 andr ∈ R ,x∈ M2 withrRx=0 , butx ∉ Q ,r∉(Q:M2) impliesrM1=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 m-systems.
Definition 2.1
Let
Proposition 2.2.
Let
The following proposition offers several characterizations of a weakly m-system
Proposition 2.3.
Let
-
(1)
P is weakly prime; -
(2)
S is a weakly m-system; -
(3) for each left ideal
, and for every submodule , if , and then ; -
(4) for each ideal
, and for every , if , and , then ; -
(5) for each
a∈ R , and for each, if , and , then
Proposition 2.4.
Let
Next we need a generalization of the notion of
Definition 2.5.
Let
Theorem 2.6.
Let
3. Weakly 2-absorbing 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 2-absorbing submodule is weakly 2-absorbing 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:RM) ; -
(2)
am ∉ N andab ∉ (K:RM) ; -
(3)
am ∉ K andab ∉ (N:RM) ; -
(4)
am ∉ K andab ∉ (K:RM) .
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:RM)m=b(N:RM)m=0. -
(2)
a(N:RM)N=b(N:RM)N=(N:RM)bN=(N:RM)bm=(N:RM)2m=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 2-absorbing 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 non-commutative 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 2-absorbing Submodules of Product Modules
Proposition 5.1.
Let
-
(1)
N1 is a 2-absorbing submodule ofM1 ; -
(2)
N1 × M2 is a 2-absorbing submodule ofM1 × M2; -
(3)
N1 × M2 is a weakly 2-absorbing submodule ofM1 × M2.
Proposition 5.2.
Let
-
(1)
N2 is a 2-absorbing submodule ofM1 ; -
(2)
M1 × N2 is a 2-absorbing submodule ofM1 × M2; -
(3)
M1 × N2 is a weakly 2-absorbing submodule ofM1 × M2 .
Proposition 5.3.
Let
Proposition 5.4.
Let
Proposition 5.5.
Let
If
-
(1)
N1 is a weakly 2-absorbing submodule ofM1 , -
(2)
N2 is a weakly 2-absorbing submodule ofM2 .
-
(1) Suppose that
is a weakly 2-absorbing submodule of . Let and such that . Clearly, for any Hence Since is a weakly 2-absorbing submodule of , or or Consequently or or . Hence N1 is a weakly 2-absorbing submodule ofM1 . -
(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)
N1 △ M2 is a weakly 2-absorbing submodule ofM1 △ M2; -
(2)
-
(a)
N1 is a weakly 2-absorbing submodule ofM1; -
(b) For each
a1,a2 ∈ R andm ∈ M1 such thata1Ra2Rm=0 ifa1a2∉(N1:M1) anda1m ∉ N1 anda2m ∉ N1 thena1Ra2M2=0.
-
(a) Suppose
(b) Let
(2)
Let
Proposition 5.8.
Let
References
- A. E. Ashour and M. Hamoda. Weakly primary submodules over non-commutative rings, J. Progr. Res. Math. 7(2016), 917-927.
- S. E. Atani and F. Farzalipour. On weakly prime submodules, Tamkang J. Math. 38(2007), 247-252.
- A. Badawi. On 2-absorbing ideals of commutative rings, Bull. Aust. Math. Soc. 75(2007), 417-429.
- A. Badawi and A. Y. Darani. On weakly 2-absorbing ideals of commutative rings, Houston J. Math. 39(2013), 441-452.
- M. Behboodi. On the prime radical and Baer's lower nilradical of modules, Acta Math. Hungar. 122(2009), 293-306.
- A. Y. Darani, F. Soheilnia, U. Tekir, and G. Ulucak. On weakly 2-absorbing primary submodules of modules over commutative rings, J. Korean Math. Soc. 54(5)(2017), 1505-1519.
- A. Y. Darani and F. Soheilnia. 2-absorbing and weakly 2-absorbing submodules, Thai J. Math. 9(2011), 577-584.
- J. Dauns. Prime modules, J. Reine Angew. Math. 298(1978), 156-181.
- N. J. Groenewald. On 2-absorbing ideals of non-commutative rings, JP J. Algebra Number Theory Appl. 40(5)(2018), 855-867.
- N. J. Groenewald. On weakly and strongly 2-absorbing ideals of non-commutative rings, submitted.
- N. J. Groenewald and B. T. Nguyen. On 2-absorbing modules over noncommutative rings, Int. Electron. J. Algebra 25(2019), 212-223.
- A. A. Tuganbaev. Multiplication modules over noncommutative rings, Sb. Math. 194(12)(2003), 1837-1864.