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 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.

Highlights

  • Introduction e FundamentalHomomorphism eorem provided by Noether [1] in 1927 shows that every homomorphism gives rise to an isomorphism and that quotient groups are merely constructions of homomorphic images

  • Since Zassenhaus lemma is used to prove the Jordan–Holder theorem and it is the generalization the Second Isomorphism eorem for groups, as a generalization, we extend Zassenhaus lemma to rings, R-modules, vector spaces, and R-algebras in terms of algebraic ideal in Section 3, i.e., give the versions of Zassenhaus lemma for rings, R-modules, and R-algebras, respectively

  • In 1934, Zassenhaus gave Zassenhaus lemma which is a generalization of the Second Isomorphism eorem for groups

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Summary

Introduction

Introduction e FundamentalHomomorphism eorem (or the First Isomorphism eorem) provided by Noether [1] in 1927 shows that every homomorphism gives rise to an isomorphism and that quotient groups are merely constructions of homomorphic images. We consider Zassenhaus lemma for rings (see eorem 5), R-modules (see eorem 6), and R-algebras (see eorem 7), respectively, and obtain the following results: (1) If R1, I1, R2, and I2 are subrings of a ring R such that Ii is an ideal of Ri for i 1, 2, 2. Goursat’s Lemma for Groups, Rings, R-Modules, and R-Algebras

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