Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2019; 56(1): 57-71

Online first article July 3, 2018      Printed January 1, 2019

https://doi.org/10.4134/BKMS.b180094

Copyright © The Korean Mathematical Society.

On clean and nil clean elements in skew t.u.p. monoid rings

Ebrahim Hashemi, Marzieh Yazdanfar

Shahrood University of Technology; Shahrood University of Technology

Abstract

Let $R$ be an associative ring with identity, $M$ a t.u.p.~monoid with only one unit and $\omega : M \rightarrow {\rm End}(R)$ a monoid homomorphism. Let $R$ be a reversible, $M$-compatible ring and $\alpha=a_{1}g_{1}+\cdots+a_{n}g_{n}$ a non-zero element in skew monoid ring $R \ast M$. It is proved that if there exists a non-zero element $\beta=b_{1}h_{1}+\cdots+b_{m}h_{m}$ in $R \ast M$ with $\alpha\beta=c$ is a constant, then there exist $1 \leq i_{0} \leq n$, $1 \leq j_{0} \leq m$ such that $g_{i_{0}}=e=h_{j_{0}}$ and $a_{i_{0}}b_{j_{0}}=c$ and there exist elements $a, 0\neq r$ in $R$ with $\alpha r=ca$. As a consequence, it is proved that $\alpha \in R \ast M$ is unit if and only if there exists $1 \leq i_{0} \leq n$ such that $g_{i_{0}}=e, a_{i_{0}}$ is unit and $a_{j}$ is nilpotent for each $j \neq i_{0}$, where $R$ is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of $R$ and those elements in skew monoid ring $R \ast M$, where $R$ is a reversible or right duo ring.

Keywords: skew monoid rings, idempotent elements, unit elements, clean elements, nil clean elements

MSC numbers: 16U60, 06F05, 16S34, 16S35

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