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On annihilations of ideals in skew monoid rings

Ahmad Moussavi|Rasul Mohammadi|Masoome Zahiri
대한수학회 (The Korean Mathematical Society)|2016.03|대한수학회지 (Journal of the KMS)|제53권 제2호|pp. 381-401

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참고문헌

키워드

McCoy ringskew monoid ringstrongly right $AB$ ringnil-reversible ringCN ringrings with Property $(A)$zip ring
주제분류수학

초록

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 $(A)$.

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 $(A)$.

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