Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respec...Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetric α-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α（1） = 1, then R is a symmetric α-ring if and only if R[x]/（x^n） is a symmetric q^--ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q（R） the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q（R） is a symmetric α-ring. Among others we also show that if a ring R is weakly 2-primal and （α, δ）-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric α^--ring.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11101217)the Natural Science Foundation of Jiangsu Province(Grant No.BK20141476)
文摘Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetric α-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α（1） = 1, then R is a symmetric α-ring if and only if R[x]/（x^n） is a symmetric q^--ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q（R） the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q（R） is a symmetric α-ring. Among others we also show that if a ring R is weakly 2-primal and （α, δ）-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric α^--ring.
基金Supported by the Provincial Natural Science Research Program of Higher Education Institution of Anhui Province of China(KJ2012Z028)Partially Supported by the Foundation for Young Talents in College of Anhui Province of China(2012SQRL039)the National Natural Science Foundation of China(11226056)
