On Abelian Rings

Authors: NAZIM AGAYEV, ABDULLAH HARMANCI, SAİT HALICIOĞLU

Abstract: 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.

Keywords: \alpha-reduced rings, \alpha-symmetric rings, \alpha-semicommutative rings, \alpha-Armendariz rings, \alpha-abelian rings

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