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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2021; 61(1): 33-48

Published online March 31, 2021

### 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 R be a non-commutative ring with identity. We define the notions of a weakly prime and a weakly 2-absorbing submodules of R and show that in the case that R commutative, the definition of a weakly 2-absorbing submodule coincides with the original definition of A. Darani and F. Soheilnia. We give an example to show that in general these two notions are different. The notion of a weakly m-system is introduced and the weakly prime radical is characterized interms of weakly m-systems. Many properties of weakly prime submodules and weakly 2-absorbing submodules are proved which are similar to the results for commutative rings. Amongst these results we show that for a proper submodule $Ni$ of an $Ri$-module $Mi$, for $i=1,2$, if $N1×N2$ is a weakly 2-absorbing submodule of $M1×M2$, then $Ni$ is a weakly 2-absorbing submodule of $Mi$ for $i=1,2$

Keywords: 2-absorbing submodule, weakly 2-absorbing submodule, prime submodule, weakly prime submodule, weakly prime radical.