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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.
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참고문헌 (14)
참고문헌 신청
A. Alhevaz / 2014 / McCoy property of skew Laurent polynomials and power series rings / J. Algebra Appl 13(2):23
D. D. Anderson / 2002 / Commutative rings whose elements are a sum of a unit and idempotent / Comm. Algebra 30(7):3327~3336
G. F. Birkenmeier / 1993 / Ring theory / World Sci. Publ. :102~129
G. F. Birkenmeier / 2003 / Triangular matrix representations of ring extensions / J. Algebra 265(2):457~477
W. Chen / 2006 / Units in polynomial rings over 2-primal rings / Southeast Asian Bull. Math 30(6):1049~1053
D. D. Anderson / 2002 / Commutative rings whose elements are a sum of a unit and idempotent / Comm. Algebra 30(7):3327~3336
G. F. Birkenmeier / 1993 / Ring theory / World Sci. Publ. :102~129
G. F. Birkenmeier / 2003 / Triangular matrix representations of ring extensions / J. Algebra 265(2):457~477
W. Chen / 2006 / Units in polynomial rings over 2-primal rings / Southeast Asian Bull. Math 30(6):1049~1053
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