$S$-$Q$-ideals and Almost $S$-$Q$-ideals of commutative rings
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Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. This article is motivated by the results of Mimouni on $Q$-ideals. In this paper, we introduce and investigate the notion of $S$-$Q$-ideals which is a generalization of $Q$-ideals introduced by Mimouni in \cite{mimouni}. Let $I\subseteq Q$ be two ideals of $R$ with $Q\cap S=\emptyset$. The ideal $I$ is said to be an $S$-$Q$-ideal if there exists $s\in S$ such that whenever $ab\in I$ with $a,\ b\in R$, then $sa\in I$ or $sb\in Q$. In Section 2, we study some properties of $S$-$Q$-ideals. Among other results, we give some characterization of the $S$-$Q$-ideal of the idealization of a module. On the other hand, we show $I$ is an $S$-$Q$-ideal of $R$ if and only if $I[X]$ is an $S$-$Q[X]$-ideal of $R[X]$, provided that $R$ is a domain and $S$ is anti-Archimedean. In Section 3, we introduce a weak version of $S$-$Q$-ideals and we study some of their properties.
Abstract
Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. This article is motivated by the results of Mimouni on $Q$-ideals. In this paper, we introduce and investigate the notion of $S$-$Q$-ideals which is a generalization of $Q$-ideals introduced by Mimouni in \cite{mimouni}. Let $I\subseteq Q$ be two ideals of $R$ with $Q\cap S=\emptyset$. The ideal $I$ is said to be an $S$-$Q$-ideal if there exists $s\in S$ such that whenever $ab\in I$ with $a,\ b\in R$, then $sa\in I$ or $sb\in Q$. In Section 2, we study some properties of $S$-$Q$-ideals. Among other results, we give some characterization of the $S$-$Q$-ideal of the idealization of a module. On the other hand, we show $I$ is an $S$-$Q$-ideal of $R$ if and only if $I[X]$ is an $S$-$Q[X]$-ideal of $R[X]$, provided that $R$ is a domain and $S$ is anti-Archimedean. In Section 3, we introduce a weak version of $S$-$Q$-ideals and we study some of their properties.