$r$-ideals in commutative rings

Abstract

In this article we introduce the concept of $r$-ideals in commutative rings (note: an ideal $I$ of a ring $R$ is called $r$-ideal, if $ab\in I$ and ${\rm Ann}(a)=(0)$ imply that $b\in I$ for each $a,b\in R$). We study and investigate the behavior of $r$-ideals and compare them with other classical ideals, such as prime and maximal ideals. We also show that some known ideals such as $z^\circ$-ideals are $r$-ideals. It is observed that if $I$ is an $r$-ideal, then so too is a minimal prime ideal of $I$. We naturally extend the celebrated results such as Cohen's theorem for prime ideals and the Prime Avoidance Lemma to $r$-ideals. Consequently, we obtain interesting new facts related to the Prime Avoidance Lemma. It is also shown that $R$ satisfies property $A$ (note: a ring $R$ satisfies property $A$ if each finitely generated ideal consisting entirely of zerodivisors has a nonzero annihilator) if and only if for every $r$-ideal $I$ of $R$, $I[x]$ is an $r$-ideal in $R[x]$. Using this concept in the context of $C(X)$, we show that every $r$-ideal is a $z^\circ$-ideal if and only if $X$ is a $\partial$-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior). Finally, we observe that, although the socle of $C(X)$ is never a prime ideal in $C(X)$, the socle of any reduced ring is always an $r$-ideal.