Classication of Units of Five Radical Zero Completely Primary Finite Rings with Variant Orders of Second Galois Ring Module Generators

Abstract

Let \(R\) be a commutative completely primary finite ring with a unique maximal ideal \(Z(R)\) such that \((Z(R))^{5}=(0) ;(Z(R))^{4} \neq(0)\). Then \(R / Z(R) \cong G F\left(p^{r}\right)\) is a finite field of order \(p^{r}\). Let \(R_{0}=G R\left(p^{k r}, p^{k}\right)\) be a Galois ring of order \(p^{k r}\) and of characteristic \(p^{k}\) for some prime number \(p\) and positive integers \(k, r\) so that \(R=R_{0} \oplus U \bigoplus V \bigoplus W \bigoplus Y\), where \(U, V, W\) and \(Y\) are \(R_{0} / p R_{0}\) - spaces considered as \(R_{0}\) modules generated by \(e, f, g\) and \(h\) elements respectively. Then \(R\) is of characteristic \(p^{k}\) where \(1 \leq k \leq 5\). In this paper, we investigate and determine the structures of the unit groups of some classes of commutative completely primary finite ring \(R\) with \(p u_{i}=p^{\xi} v_{j}=p w_{k}=p y_{l}=0\), where \(\xi=2,3 ; 1 \leq i \leq e, 1 \leq j \leq f, 1 \leq k \leq g\), and \(1 \leq l \leq h\).