The symmetric and asymmetric version of Goursat's Lemma

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 $1\le i\le 3$ 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 $M=\tilde{N}\times N_n$ and $M=N_1\times \cdots \times N_n$. Further, we extended similar conclusions to $R$-algebras (rings as corollary).