Classification of Unit Groups of Five Radical Zero Completely Primary Finite Rings Whose First and Second Galois Ring Module Generators Are of the Order p k , k = 2 , 3 , 4

Abstract

Let $R_0=GR\left(p^{kr},p^k\right)$ be a Galois maximal subring of $R$ so that $R=R_0\oplus U\oplus V\oplus W\oplus Y$ , where $U,V,W$ , and $Y$ are $R_0/{pR}_0$ spaces considered as $R_0$ -modules, generated by the sets $\left\{u_1,\cdots ,u_e\right\},\left\{v_1,\cdots ,v_f\right\},\left\{w_1,\cdots ,w_g\right\}$ , and $\left\{y_1,\cdots ,y_h\right\}$ , respectively. Then, $R$ is a completely primary finite ring with a Jacobson radical $Z\left(R\right)$ such that ${\left(Z\left(R\right)\right)}^5=\left(0\right)$ and ${\left(Z\left(R\right)\right)}^4\ne \left(0\right)$ . The residue field $R/Z\left(R\right)$ is a finite field $GF\left(p^r\right)$ for some prime $p$ and positive integer $r$ . The characteristic of $R$ is $p^k$ , where $k$ is an integer such that $1\le k\le 5$ . In this paper, we study the structures of the unit groups of a commutative completely primary finite ring $R$ with $p^{\psi }{u}_i=0$ , $\psi =2,3,4$ ; $p^{\zeta }{v}_j=0$ , $\zeta =2,3$ ; ${pw}_k=0$ , and ${py}_l=0$ ; $1\le i\le e$ , $1\le j\le f$ , $1\le k\le g$ , and $1\le l\le h$ .