Iranian Journal of

Let X be a topological space and R be a subring of R^X. By determining some special topologies on X associated with
the subring R, characterizations of maximal fixxed and maximal growing ideals in R of the form M_x(R) are given. Moreover, the
classes of z_R-ideals and z⁰_R-ideals are introduced in R which are topological generalizations of z-ideals and z⁰-ideals of C(X), respectively. Various characterizations of these ideals are established, also, coincidence of z_R-ideals with z-ideals and z_R-ideals with z-ideals in R are investigated. It turns out that some fundamental statements in the context of C(X) are extended to the subrings of R^X