Generalization of Goursat's Theorem for Subrings of Direct Products of n Rings
Abstract
Bauer et al. describe Goursat’s theorem, representing the characteristics of subgroups of a direct product of two or more groups. In this paper, we expand into a ring structure that describes the characteristics of subrings of a direct product of rings. This research method is to analogize the evidence by Bauer et al. in the group for generalization. In our main results, every subring of the direct product of rings is determined by ring epimorphism between the ring and factor ring.
Keywords: Goursat’s theorem, subrings, rings
References
[1] Goursat E. “Sur les substitutions orthogonales et les divisions régulières de l’espace,” Annales scientifiques de l’École normale supérieure. vol. 6, p. 1889.
[2] Anderson DD, Camillo V. “Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat’s lemma,” Presented at the (2009). https://doi.org/10.1090/conm/480/09364.
[3] Tóth L. “Subgroups of finite abelian groups having rank two via Goursat’s Lemma,” Tatra Mountains Mathematical Publications. vol. 59, no. 1, p. 2014.
[4] L. Tóth, “Counting subrings of the ring Zm × Zn.,” Journal of the Korean Mathematical Society. vol. 56, no. 6, p. 2019.
[5] Oh JM. “An explicit formula for the number of subgroups of a finite abelian p-group up to rank 3.,” Communications of the Korean Mathematical Society. vol. 28, no. 4, p. 2013. https://doi.org/10.4134/CKMS.2013.28.4.649.
[6] Dickson SE. On algebras of finite representation type. Trans Am Math Soc. 1969;135(0):127–41.
[7] F. Meng, J. Guo, and L. Guo, “On the extensions of Zassenhaus Lemma and Goursat’s Lemma to algebraic structures,.” Journal of Mathematics. vol. 2022, p. 2022. https://doi.org/10.1155/2022/7705500.
[8] B.R.A. Mbarga, “Homomorphic relations and Goursat Lemma.,” Earthline Journal of Mathematical Sciences. p. 2022. https://doi.org/10.34198/ejms.10122.169181.
[9] Petrillo J. “Student research projects Goursat’s other theorem,” The College Mathematics Journal. vol. 40, no. 2, p. 2010.
[10] Kublik K. Generalizations of Goursat’s theorem for groups. Rose-Hulman Undergraduate Mathematics Journal. 2010;11(1):6.
[11] M. Hampejs, N. Holighaus, L. Tóth, and C. Wiesmeyr, “Representing and counting the subgroups of the group Zm × Zn.,” Journal of Numbers. vol. 2014, p. 2014. https://doi.org/10.1155/2014/491428.
[12] Bauer K, Sen D, Zvengrowski P. “A generalized Goursat lemma.,” Tatra Mountains Mathematical Publications. vol. 64, no. 1, p. 2015. https://doi.org/10.1515/tmmp-2015- 0039.
[13] D. neuen and P. Schweitzer, “Subgroups of 3-factor direct products.,” Tatra Mountains Mathematical Publications. vol. 73, no. 1, p. 2019.
[14] B. Masterson and G. Pfeiffer, “On the table of marks of a direct product of finite groups.,” Journal of Algebra. vol. 499, p. 2018. https://doi.org/10.1016/j.jalgebra.2017.12.019.
[15] B.R.A. Mbarga, “Decomposition of Goursat matrices and subgroups of Zm x Zn.,” Earthline Journal of Mathematical Sciences. p. 2021. https://doi.org/10.34198/ejms.6221.439454.
[16] Mbarga BR. “Some remarks on Goursat lemma.,” Algebraic Structures and their Applications. vol. 8, no. 2, p. 2021.