On Abelian Rings

Let α be an endomorphism of an arbitrary ring R with identity. In this note, we introduce the notion of α-abelian rings which generalizes abelian rings. We prove that α-reduced rings, α-symmetric rings, α-semicommutative rings and α-Armendariz rings are α-abelian. For a right principally projective ring R, we also prove that R is α-reduced if and only if R is α-symmetric if and only if R is α-semicommutative if and only if R is α-Armendariz if and only if R is α-Armendariz of power series type if and only if R is α-abelian.