On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures

Abstract

The Jordan–Hölder theorem is proved by using Zassenhaus lemma which is a generalization of the Second Isomorphism Theorem for groups. Goursat’s lemma is a generalization of Zassenhaus lemma, it is an algebraic theorem for characterizing subgroups of the direct product of two groups $G_1\times G_2$ , and it involves isomorphisms between quotient groups of subgroups of $G_1$ and $G_2$ . In this paper, we first extend Goursat’s lemma to $R$ -algebras, i.e., give the version of Goursat’s lemma for algebras, and then generalize Zassenhaus lemma to rings, $R$ -modules, and $R$ -algebras by using the corresponding Goursat’s lemma, i.e., give the versions of Zassenhaus lemma for rings, $R$ -modules, and $R$ -algebras, respectively.