J. Korean Math. Soc. 2016; 53(2): 381-401
Printed March 1, 2016
https://doi.org/10.4134/JKMS.2016.53.2.381
Copyright © The Korean Mathematical Society.
Rasul Mohammadi, Ahmad Moussavi, and Masoome Zahiri
Tarbiat Modares University, Tarbiat Modares University, Tarbiat Modares University
According to Jacobson \cite{Jacobson}, a right ideal is bounded if it contains a non-zero ideal, and Faith \cite{Faith2} called a ring strongly right bounded if every non-zero right ideal is bounded. From \cite{Hwang}, a ring is strongly right $AB$ if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property ($A$) and the conditions asked by Nielsen \cite{Nielsen}. It is shown that for a u.p.-monoid $M$ and $\sigma: M \rightarrow {\rm End}(R)$ a compatible monoid homomorphism, if $R$ is reversible, then the skew monoid ring $R\ast M$ is strongly right $AB$. If $R$ is a strongly right $AB$ ring, $M$ is a u.p.-monoid and $\sigma: M \rightarrow {\rm End}(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring $R\ast M$ has right Property $(A)$.
Keywords: skew monoid ring, McCoy ring, strongly right $AB$ ring, nil-reversible ring, CN ring, rings with Property $(A)$, zip ring
MSC numbers: 16D25, 16D70, 16S34
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