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

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 G1×G2, and it involves isomorphisms between quotient groups of subgroups of G1 and G2. 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.