Weakly 2-Absorbing ideals In Non-Commutative Rings
Let R be a commutative ring with identity. Various generalizations of prime ideals and 2-absorbing ideals have been studied. For example, a proper ideal I of R is weakly (resp., almost prime ideal) if a, b ∈ R with 0 6= ab ∈ I (resp., ab ∈ I −I 2 ), then either a ∈ I or b ∈ I. A proper ideal I of R is weakly 2-absorbing (resp., almost 2-absorbing ideal) if a, b, c ∈ R with 0 6= abc ∈ I (resp., abc ∈ I − I 2 ), then either ab ∈ I or ac ∈ I, or bc ∈ I. In this paper we only consider non-commutative rings and introduce weakly 2-absorbing ideals and show that it enjoys analogs of many of the properties of 2-absorbing ideals. Also we classify all rings by investigating the conditions that need to be imposed on the ring for which every proper 2-absorbing ideal is weakly 2-absorbing ideal. However, it is not possible that every ideal of a ring is weakly 2-absorbing, but if the ring R3 = 0, then every proper ideal of R is weakly 2-absorbing by Theorem 2.2. We introduce almost 2-absorbing ideal and its relationship with 2-absorbing ideal and weakly 2-absorbing ideal in non-commutative rings. After studying and comparing the two algebraic structures (commutative and non-commutative), several properties and characteristics hold in commutative case and fail to hold in non-commutative case.
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