Affiliation:
1. Department of Mathematics, College of Sciences, King Saud University, Riyadh 11451, Saudi Arabia
Abstract
Let R be a finite commutative chain ring with invariants p,n,r,k,m. The purpose of this article is to study j-diagrams for the one group H=1+J(R) of R, where J(R)=(π) is Jacobson radical of R. In particular, we prove the existence and uniqueness of j-diagrams for such one group. These j-diagrams help us to solve several problems related to chain rings such as the structure of their unit groups and a group of all symmetries of {πk′}, where k′∣k. The invariants p,n,r,k,m and the Eisenstein polynomial by which R is constructed over its Galois subring determine fully the j-diagram for H.
Funder
King Saud University, Riyadh, Saudi Arabia
Subject
Physics and Astronomy (miscellaneous),General Mathematics,Chemistry (miscellaneous),Computer Science (miscellaneous)
Reference18 articles.
1. Alabiad, S., and Alkhamees, Y. (2021). Recapturing the structure of group of units of any finite commutative chain rings. Symmetry, 13.
2. On the group of units for certain rings;Ayoub;J. Number Theory,1972
3. On diagrams for abelian groups;Ayoub;J. Number Theory,1970
4. Alabiad, S., and Alkhamees, Y. (2021). On automorphism groups of finite chain rings. Symmetry, 13.
5. Finite commutative chain rings;Hou;Finite Fields Appl.,2001