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, Ix is an r-ideal in Rx. 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.