Journal of the
Korean Mathematical Society
JKMS

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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.

On annihilations of ideals in skew monoid rings

Rasul Mohammadi, Ahmad Moussavi, and Masoome Zahiri

Tarbiat Modares University, Tarbiat Modares University, Tarbiat Modares University

Abstract

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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