r-ideals in commutative rings

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∈ I and Ann(a)=(0) imply that b∈ I for each a,b∈ 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^∘-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^∘-ideal if and only if X is a ∂-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.