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# A Theorem on Unit-Regular Rings

Published:2009-12-04
Printed: Jun 2010
• Tsiu-Kwen Lee
• Yiqiang Zhou
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## Abstract

Let $R$ be a unit-regular ring and let $\sigma$ be an endomorphism of $R$ such that $\sigma (e)=e$ for all $e^2=e\in R$ and let $n\ge 0$. It is proved that every element of $R[x \mathinner;\sigma]/(x^{n+1})$ is equivalent to an element of the form $e_0+e_1x+\dots +e_nx^n$, where the $e_i$ are orthogonal idempotents of $R$. As an application, it is proved that $R[x \mathinner; \sigma ]/(x^{n+1})$ is left morphic for each $n\ge 0$.
 Keywords: morphic rings, unit-regular rings, skew polynomial rings
 MSC Classifications: 16E50 - von Neumann regular rings and generalizations 16U99 - None of the above, but in this section 16S70 - Extensions of rings by ideals 16S35 - Twisted and skew group rings, crossed products