具有对称自同态的环及其扩张

Rings with Symmetric Endomorphisms and Their Extensions

中文关键词:  对称环  弱对称σ-环  多项式扩张  典范商环扩张  Ore扩张

英文关键词:symmetric $\alpha$-ring  weak symmetric $\alpha$-ring  polynomial extension  classical quotient ring extension  Ore extension

中文摘要:

      设 $R$ 是一个具有环自同态 $\alpha$ 和 $\alphaα$-导子 $\delta$ 的环. 本文分别引进对称 $\alphaα$-环和弱对称 $\alphaα$-环的概念, 拓展对称环和弱对称环的研究. 讨论对称$\alphaα$-环与相关环的关系,研究对称 $\alphaα$-环和弱对称 $\alphaα$-环的扩张性质. 特别地, 我们证明了:(1)如果 $R$ 是约化环且 $\alphaα(1)=1$, 则 $R$ 是对称 $\alphaα$-环当且仅当 $R[x]/(x^{n})$ 是对称 $\alphaα$-环;(2)设 $R$ 是右 Ore 环, $Q(R)$ 是 $R$ 的典范右商环, $\alphaα$ 是环 $R$ 的一个自同构, 则 $R$ 是对称 $\alphaα$-环当且仅当 $Q(R)$ 是对称 $\alphaα$-环; (3)如果 $R$ 是弱 $2$-素的 $(\alpha,\delta)$-可压缩环, 则 $R$ 是弱对称 $\alphaα$-环当且仅当 $R$ 的 Ore 扩张 $R[x; \alpha,\delta]$ 是弱对称 $\alphaα$- 环.

英文摘要:

      Let $R$ be a ring with an endomorphism $\alpha$ and an $\alpha$-derivation $\delta$. We introduce the notions of symmetric $\alpha$-rings and weak symmetric $\alpha$-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetric $\alpha$-rings and related rings and investigate their extensions. We prove that if $R$ is a reduced ring and $\alpha(1)=1$, then $R$ is a symmetric $\alpha$-ring if and only if $R[x]/(x^{n})$ is a symmetric $\bar{\alpha}$-ring for any positive integer $n$. Moreover, it is proven that if $R$ is a right Ore ring, $\alpha$ an automorphism of $R$ and $Q(R)$ the classical right quotient ring of $R$, then $R$ is a symmetric $\alpha$-ring if and only if $Q(R)$ is a symmetric $\bar{\alpha}$-ring. Among others we also show that if a ring $R$ is weakly $2$-primal and $(\alpha,\delta)$-compatible, then $R$ is a weak symmetric $\alpha$-ring if and only if the Ore extension $R[x;\alpha,\delta]$ of $R$ is a weak symmetric $\bar{\alpha}$-ring.