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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 × 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.

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