On The Extensions Of Zassenhaus Lemma And Goursat’s Lemma To Algebraic Structures

The Jordan-H（?）lder 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 charac-terizing subgroups of the direct product of two groups G₁×G₂,and it involves isomorphisms between quotient groups of subgroups of G₁and 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.