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It is proved that if a ring R is semiabelian, then so is the skew polynomial ring R[x;σ], where σ is an endomorphism of R satisfying σ(e)=e for all e∈E(R). Some characterizations and properties of semiabelian rings are studied.

We introduce in this paper the concept of left 𝑊𝑀𝐶2 rings and concern ourselves with rings containing an injective maximal left ideal. Some known results for left idempotent reflexive rings and left 𝐻𝐼 rings can be extended to left 𝑊𝑀𝐶2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with nonzero socle and extend some known results on strongly regular rings.