Some results on the intermediate rings of complex-valued continuous functions

Let X be a Tychonoff space and C(X), C ( X , ℂ ) be the rings of all real-valued and complex-valued continuous functions defined on X respectively. For each intermediate subring A(X) of C(X), Acharyya and De have introduced the notion of zβA -ideals in A(X) and zβA-filters on β X. For each A(X), we extend this notion to zβ[A(X)]c -ideals and zβ[A(X)]c-filters for an intermediate subring [A(X)]c of C ( X , ℂ ). We establish a correspondence between the collection of zβ[A(X)]c-ideals in the subrings [A(X)]c of C ( X , ℂ ) and the collection of zβ[A(X)]c-filters on β X. We study the properties of the zβ[A(X)]c-ideals. We also deduce that the structure space of the subrings [A(X)]c is homeomorphic to β X.