Summary: Let \(A\) be an Archimedean \(f\)-ring with identity and assume that \(A\) is equipped with another multiplication \(\ast\) so that \(A\) is an \(f\)-ring with identity \(u\). Obviously, if \(\ast\) coincides with the original multiplication of \(A\) then \(u\) is idempotent in \(A\) (i.e., \(u^2=u\)). Conrad proved that the converse also holds, meaning that, it suffices to have \(u^2=u\) to conclude that \(\ast\) equals the original multiplication on \(A\). The main purpose of this paper is to extend this result as follows. Let \(A\) be a (not necessary unital) Archimedean \(f\)-ring and \(B\) be an \(\ell\)-subgroup of the underlaying \(\ell\)-group of \(A\). We will prove that if \(B\) is an \(f\)-ring with identity \(u\), then the equality \(u^2=u\) is a necessary and sufficient condition for \(B\) to be an \(f\)-subring of \(A\). As a key step in the proof of this generalization, we will show that the set of all \(f\)-subrings of \(A\) with the same identity has a smallest element and a greatest element with respect to the inclusion ordering. Also, we shall apply our main result to obtain a well known characterization of \(f\)-ring homomorphisms in terms of idempotent elements.