Remarks on intermediate c-rings of c(x)

Abstract

Let X be a Tychonoff space and A(X) be a subring of C(X) containing C*(X). We introduce the notion of z^v_a-ideal in A(X). It is observed that the class of z^v_a-ideals contains the class of z_a-ideals and is contained in the class of z-ideals of A(X). These containments may be proper. It turns out that coincidence of z-ideals of A(X) with z^v_a -ideals characterizes intermediate C-rings of C(X).

Mathematics Subject Classification (2010): 54C30, 46E25.

Keywords: z-ideal, z_a-ideal, z^v_a-ideals, intermediate ring, intermediate C-ring