@article{Lai2025, 
author = {Kailing Lai and Fanning Meng and Leiling Zhang},
title = {The symmetric and asymmetric version of Goursat's Lemma},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {11},
pages = {6952-6970},
keywords = {Goursat's lemma, R-modules, R-algebras, direct product},
url = {https://www.sciopen.com/article/10.3934/era.2025306},
doi = {10.3934/era.2025306},
abstract = {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    ×      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        π    i    (  H  )  =      G    i  , the injectivity        ρ    i  , and        H          i      22        =      H          i      12        ∩      H          i      21       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        M    1    ×  ⋯  ×      M    n   has the form    M  =            N      ~        ×      N    n   and    M  =      N    1    ×  ⋯  ×      N    n  . Further, we extended similar conclusions to    R-algebras (rings as corollary).}
}