AI Summary & Key Takeaways
AI InsightFor 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.
📌 3 Key Takeaways
Abstract
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.
Özet
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.