LAUSR

Abstract In F. Azarpanah et al. (2008) [3] the authors have given some algebraic properties of the ring C ( X ) / C F ( X ) , where C F ( X ) = O β X ∖ I ( X ) . In this paper, first, we show that C ( X ) / C F ( 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 ) / O A and C ( X ) / M A whenever A ⊆ β X (or just a closed one, in some cases) and then topological conditions on A for which every prime (maximal) ideal of C ( X ) / O A (resp., C ( X ) / M A ) 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 ) / O A is an EIN -ring if and only if C ( X ) / M A 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 ) / O A (resp., C ( X ) / M A ) are characterized. We prove also that dim ( C ( X ) / O A ) ≥ dim C ( X ) / M A , where dim C ( X ) / M A denotes the Goldie dimension of C ( X ) / M A , and the inequality may be strict.