The symmetric and asymmetric version of Goursat’s Lemma

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Abstract

Goursat’s lemma gives a good method to describe subgroups of the direct product of two groups G₁,G₂, and to determine whether subgroups of G₁ ×G₂ are direct products. However, the usual symmetric version of Goursat’s lemma is difficult to describe subgroups of a direct product of a finite number of groups. Fortunately, the asymmetric version of Goursat’s lemma provide a new method to solve the difficulty. In this paper, we used additional conditions π_i(H) =G_i, the injectivity ρ_i, and H_i22 =H_i12 ∩ H_i21 for 1 ≤ i ≤ 3 to give some related results about groups (resp.R-modules,R-algebras (rings as corollary)), and then we gave the answer on whether a R-submodule M of M₁ × ...×M_n has the form $M=\widetilde{N} \times N_{n}$ and M =N₁ × ...×N_n. Further, we extended similar conclusions to R-algebras (rings as corollary).

Keywords

Goursat’s lemma / R-modules / R-algebras / direct product

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