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.
\alpha-reduced rings, \alpha-symmetric rings, \alpha-semicommutative rings, \alpha-Armendariz rings, \alpha-abelian rings
AGAYEV, NAZIM; HARMANCI, ABDULLAH; and HALICIOĞLU, SAİT
"On Abelian Rings,"
Turkish Journal of Mathematics: Vol. 34:
4, Article 4.
Available at: https://journals.tubitak.gov.tr/math/vol34/iss4/4