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