Intermediate algebras in Archimedean semiprime f-algebras

We introduce the notion of bounded quasi-inversion closed semiprime f-algebras and we prove that, if A is such an algebra, then any intermediate algebra in A is an order ideal of A. This extends a recent result by Dominguez who has dealt with the unital case (the problem on C(X)-type spaces has been solved earlier by Dominguez, Gomez-Perez, and Mulero). Our results are illustrated by examples of algebras of continuous functions and algebras of measurable functions.