On Abelian Rings

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