Strongly Nil--Clean Rings
A *-ring $R$ is called a strongly nil-*-clean ring if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this article, we show that $R$ is a strongly nil-*-clean ring if and only if every idempotent in $R$ is a projection, $R$ is periodic, and $R...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
22.11.2012
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Subjects | |
Online Access | Get full text |
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Summary: | A *-ring $R$ is called a strongly nil-*-clean ring if every element of $R$ is
the sum of a projection and a nilpotent element that commute with each other.
In this article, we show that $R$ is a strongly nil-*-clean ring if and only if
every idempotent in $R$ is a projection, $R$ is periodic, and $R/J(R)$ is
Boolean. For any commutative *-ring $R$, we prove that the algebraic extension
$R[i]$ where $i^2=\mu i+\eta$ for some $\mu,\eta\in R$ is strongly nil-*-clean
if and only if $R$ is strongly nil-*-clean and $\mu\eta$ is nilpotent. The
relationships between Boolean *-rings and strongly nil-*-clean rings are also
obtained. |
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DOI: | 10.48550/arxiv.1211.5286 |