A note on the FIP property for extensions of commutative rings

Abstract

A ring extension R ⊂ S is said to be FIP if it has only finitely many intermediate rings between R and S. The main purpose of this paper is to characterize the FIP property for a ring extension, where R is not (necessarily) an integral domain and S may not be an integral domain. Precisely, we establish a generalization of the classical Primitive Element Theorem for an arbitrary ring extension. Also, various sufficient and necessary conditions are given for a ring extension to have or not to have FIP, where S = R[α] with α a nilpotent element of S.