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A Theorem on Unit-Regular Rings

Published online by Cambridge University Press:  20 November 2018

Tsiu-Kwen Lee  and
Yiqiang Zhou
Tsiu-Kwen Lee
Affiliation:
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan e-mail: tklee@math.ntu.edu.tw
Yiqiang Zhou
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St.John’s, NL A1C 5S7 e-mail: zhou@math.mun.ca
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Abstract

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Let $R$ be a unit-regular ring and let $\sigma $ be an endomorphism of $R$ such that $\sigma \left( e \right)\,=\,e$ for all ${{e}^{2}}\,=\,e\,\in \,R$ and let $n\,\ge \,0$. It is proved that every element of $R[x;\,\sigma ]/\left( {{x}^{n+1}} \right)$ is equivalent to an element of the form ${{e}_{0}}\,+\,{{e}_{1}}x\,+\,\cdots \,+\,{{e}_{n}}{{x}^{n}}$, where the ${{e}_{i}}$ are orthogonal idempotents of $R$. As an application, it is proved that $R[x;\,\sigma ]/\left( {{x}^{n+1}} \right)$ is left morphic for each $n\,\ge \,0$.

Keywords

16E5016U9916S7016S35morphic ringsunit-regular ringsskew polynomial rings
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 2 , 01 June 2010 , pp. 321 - 326
Copyright
Copyright © Canadian Mathematical Society 2010

References

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