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 ∈ 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.