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

Anahtar Kelimeler:
## a-reduced rings, a-symmetric rings, a-semicommutative rings, a-Armendariz rings, a-abelian rings

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

Keywords:
## a-reduced rings, a-symmetric rings, a-semicommutative rings, a-Armendariz rings, a-abelian rings,

**ISSN:**1300-0098**Yayın Aralığı:**6**Yayıncı:**TÜBİTAK

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