On Weakly Prime and Weakly 2-absorbing Modules over Noncommutative Rings
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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.
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.
참고문헌 (12)
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