On Regular Ideals in Reduced Rings

A R Aliabady, R Mohamadian, S Nazari

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).