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STRONGLY CLEAN POWER SERIES RINGS

Published online by Cambridge University Press:  09 February 2007

Jianlong Chen
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China (jlchen@seu.edu.cn)
Yiqiang Zhou
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada (zhou@math.mun.ca)
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Abstract

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An element $a$ in a ring $R$ with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And $a\in R$ is called strongly $\pi$-regular if both chains $aR\supseteq a^2R\supseteq\cdots$ and $Ra\supseteq Ra^2\supseteq\cdots$ terminate. A ring $R$ is called strongly clean (respectively, strongly $\pi$-regular) if every element of $R$ is strongly clean (respectively, strongly $\pi$-regular). Strongly $\pi$-regular elements of a ring are all strongly clean. Let $\sigma$ be an endomorphism of $R$. It is proved that for $\varSigma r_ix^i\in R[[x,\sigma]]$, if $r_0$ or $1-r_0$ is strongly $\pi$-regular in $R$, then $\varSigma r_ix^i$ is strongly clean in $R[[x,\sigma]]$. In particular, if $R$ is strongly $\pi$-regular, then $R[[x,\sigma]]$ is strongly clean. It is also proved that if $R$ is a strongly $\pi$-regular ring, then $R[x,\sigma]/(x^n)$ is strongly clean for all $n\ge1$ and that the group ring of a locally finite group over a strongly regular or commutative strongly $\pi$-regular ring is strongly clean.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2007