Let A be an Archimedean f-ring with identity and assume that A is equipped with another multiplication \(*\) so that A is an f-ring with identity u. Obviously, if \(*\) 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 \(*\) 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.

令A是一个具有恒等式的阿基米德f环，并假设A配备了另一个乘法\(*\)，因此A是一个具有恒等式的f环。显然，如果\(*\)与 A 的原始乘法一致，则u在A中是幂等的（即\(u^{2}=u\)）。康拉德证明了逆也成立，这意味着只要有\(u^{2}=u\)就可以得出结论\(*\)等于A上的原始乘法. 本文的主要目的是将这一结果扩展如下。令A为（非必须的单位）阿基米德f环，B为A的底层\(\ell \) -group的\(\ell \) -子群。我们将证明如果B是具有身份u的f环，那么等式\(u^{2}=u\)是B成为A的f子环的充分必要条件。作为证明这一概括的关键步骤，我们将证明 A 的所有f子环的集合对于包含排序，具有相同标识的元素具有最小元素和最大元素。此外，我们将应用我们的主要结果来获得关于幂等元素的f环同态的众所周知的表征。