Goursat's lemma gives a good method to describe subgroups of the direct product of two groups $ G_1, G_2 $, and to determine whether subgroups of $ G_1\times G_2 $ 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 $ \pi_i(H) = G_i $, the injectivity $ \rho_i $, and $ H_{i22} = H_{i12}\cap H_{i21} $ for $ 1\leq i\leq 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 $ M_1\times \cdots \times M_n $ has the form $ M = \widetilde N\times N_n $ and $ M = N_1\times \cdots \times N_n $. Further, we extended similar conclusions to $ R $-algebras (rings as corollary).
Citation: Kailing Lai, Fanning Meng, Leiling Zhang. The symmetric and asymmetric version of Goursat's Lemma[J]. Electronic Research Archive, 2025, 33(11): 6952-6970. doi: 10.3934/era.2025306
Goursat's lemma gives a good method to describe subgroups of the direct product of two groups $ G_1, G_2 $, and to determine whether subgroups of $ G_1\times G_2 $ 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 $ \pi_i(H) = G_i $, the injectivity $ \rho_i $, and $ H_{i22} = H_{i12}\cap H_{i21} $ for $ 1\leq i\leq 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 $ M_1\times \cdots \times M_n $ has the form $ M = \widetilde N\times N_n $ and $ M = N_1\times \cdots \times N_n $. Further, we extended similar conclusions to $ R $-algebras (rings as corollary).
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Kailing Lai, Fanning Meng, Leiling Zhang. The symmetric and asymmetric version of Goursat's Lemma[J]. Electronic Research Archive, 2025, 33(11): 6952-6970. doi: 10.3934/era.2025306
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