Turkish Journal of Mathematics h t t p : / / j o u r n a l s . t u b i t a k . g o v . t r / m a t hAbstract: 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 AvoidanceLemma 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. 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.

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Let R be a commutative ring with identity and X be a Tychonoff space. An ideal I of R is Von Neumann regular (briefly, regular) if for every a ∈ I, there exists b ∈ R such that a = a 2 b. In the present paper, we obtain the general form of a regular ideal in C(X) which is O A , for some closed subset A of βX, for which A c ∩ X ⊆ (P(X)) • , where P(X) is the set of all P-points of X. We show that the ideals and subrings such as C K (X), C ψ (X), C ∞ (X), Soc m C(X) and M βX\X are regular if and only if they are equal to the socle of C(X). We carry further the study of the maximal regular ideal, for instance, it is shown that for a vast class of topological spaces (we call them OPD-spaces) the maximal regular ideal is O X\I(X) , where I(X) is the set of isolated points of X. Also, for this class, the socle of C(X) is the maximal regular ideal if and only if I(X) contains no infinite closed set. We also show that C(X) contains an ideal which is both essential and regular if and only if (P(X)) • is dense in X. Finally it is shown that, for semiprimitive rings pure ideals are of the form O A which A is a closed subset of Max(R), also a P-point of X = Max(R) is introduced and it is shown that the maximal regular ideal of an arbitrary ring R is O X\P(X) , which P(X) is the set of P-points of X = Max(R).

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