Algebraic properties of some factor rings of C(X)

In F. Azarpanah et al. (2008) [3] the authors have given some algebraic properties of the ring C(X)/CF(X), where CF(X)=OβX∖I(X). In this paper, first, we show that C(X)/CF(X) is a C-ring if and only if the set of isolated points of X is finite. Next, we generalize this work for rings C(X)/OA and C(X)/MA whenever A⊆βX (or just a closed one, in some cases) and then topological conditions on A for which every prime (maximal) ideal of C(X)/OA (resp., C(X)/MA) is essential are characterized. We call a ring R an EIN-ring if for each two orthogonal ideals I, J of R which are generated by two subsets of idempotents, Ann(I)+Ann(J)=R. It is shown for a closed subset A of βX that C(X)/OA is an EIN-ring if and only if C(X)/MA is an EIN-ring if and only if A is an EF-space. Minimal ideals, socle and the intersection of all essential maximal ideals of C(X)/OA (resp., C(X)/MA) are characterized. We prove also that dim(C(X)/OA)≥dimC(X)/MA, where dimC(X)/MA denotes the Goldie dimension of C(X)/MA, and the inequality may be strict.