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

Authors: Hezron Saka Were; Maurice Owino Oduor
Journal: Journal of Mathematics
 (2022)

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## Abstract

LetR0=GRpkr,pk be a Galois maximal subring of R so that R=R0⊕U⊕V⊕W⊕Y, where U,V,W, and Y are R0/pR0 spaces considered as R0-modules, generated by the sets u1,⋯,ue,v1,⋯,vf,w1,⋯,wg, and y1,⋯,yh, respectively. Then, R is a completely primary finite ring with a Jacobson radical ZR such that ZR5=0 and ZR4≠0. The residue field R/ZR is a finite field GFpr for some prime p and positive integer r. The characteristic of R is pk, where k is an integer such that 1≤k≤5. In this paper, we study the structures of the unit groups of a commutative completely primary finite ring R with pψui=0, ψ=2,3,4; pζvj=0, ζ=2,3; pwk=0, and pyl=0; 1≤i≤e, 1≤j≤f, 1≤k≤g, and 1≤l≤h.


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## Body

## 1. Introduction

Completely primary finite rings with identity1≠0 have been an active area of research in the recent years probably because they play a crucial role in the classification of finite rings. The unit groups of completely primary finite rings with maximal ideal ZR such that ZR3=0 with ZR2≠0 have been classified by Chikunji [1–3]. Oduor and Onyango [4] constructed a class of completely primary finite rings in which ZR4=0 with ZR3≠0 and determined the structures of their group of units for all the characteristics of the ring R. Recently, Were et al. [5] gave a construction of a completely primary finite ring satisfying the conditions ZR5=0; ZR4≠0 and further determined the unit groups restricted to some conditions. This construction involved idealization of R0-modules, whose choice was based on Wilson [6]. Were et al. [7] classified the group of units of five radical zero completely primary finite rings with variant orders of second Galois ring module generators. The structure of unit groups for an abelian group have been determined by various techniques. Ayoub [8] studied groups which possess a particular type of series called j- diagram and determined the structure of the subgroups. In [9], Ayoub obtained various results based on the ideas regarding j-diagrams for the abelian p-groups. The j-diagram technique has been used by Alabiad and Alkhamees [10] to determine the structure of unit groups of a commutative chain ring where p−1k by fixing Ayoub’s approach and introduced a system of generators for the unit groups as well as enumerating the generators. In what follows, R shall denote a finite completely primary ring, ZR its maximal set of zero divisors (including zero). We shall also denote the coefficient Galois subring GRpkr,pk of characteristic pk and order pkr of the ring R by R0. For the previous related work, we refer to [2, 3, 5, 10, 11–13].The remaining part of this paper is organized as follows. In Section2, we give the construction of five radical zero completely primary finite rings whose structure of unit groups are considered in this paper. In Section 3, we investigate and determine the structure of the group of units R∗ of R for all characteristics pk,1≤k≤5, under the restrictions pψui=0, pζvj=0, pwk=0, pyl=0, where ψ=2,3,4; ξ=2,3; 1≤i≤e, 1≤j≤f, 1≤k≤g, and 1≤l≤h. Finally, in Section 4, we give the conclusion of this research and some areas which future researchers may dwell on. This is a continuation of the author’s earlier research on the classification of unit groups of five radical zero commutative completely primary finite rings.


## 2. Preliminaries

The following result due to Wilson [6] formed the basis for the choice of the ring in this work as an R0-module.LetR be a completely primary finite ring of characteristic pk with radical ZR such that R/ZR≅GFpr. Then, there exists an independent generating set b1,b2,⋯,bm of R as a left GRpkr,pr-module such that
(1)
b1=1;b2,⋯,bm∈ZR(2)
GRpkr,prbi is GRpkr,pr,GRpkr,pr-submodule of RThis paper considers a specific case of the general construction of five radical zero commutative completely primary finite rings where the automorphism is the identity inR0 and the ring R is commutative.


### 2.1. Construction

LetR0=GRpkr,pk be a Galois ring of order pkr and characteristic pk where p is a prime integer, 1≤k≤5 and r∈ℤ+. Suppose U,V,W, and Y are R0/pR0-spaces considered R0 modules generated by e,f,g, and h elements, respectively, such that the corresponding generating sets are u1,⋯,ue, v1,⋯,vf, w1,⋯,wg, and y1,⋯,yh, so that R=R0⊕U⊕V⊕W⊕Y is an additive abelian group. Then, on the additive group, we define multiplication by the following relations:
(i)
Ifk=1, then uiui′=ui′ui=vj, uivj=vjui=wk, uiwk=wkui=yl, uiyl=ylui=0, vjvj′=vj′vj=yl, vjwk=wkvj=0, vjyl=ylvj=0, wkwk′=wk′wk=0, wkyl=ylwk=0, ylyl′=yl′yl=0(ii)
Ifk=2, then uiui′=ui′ui=pr0+pui+vj, uivj=vjui=pui+wk, uiwk=wkui=pui+yl, uiyl=ylui=pui, vjvj′=vj′vj=yl, vjwk=wkvj=0, vjyl=ylvj=0, wkwk′=wk′wk=0, wkyl=ylwk=0, ylyl′=yl′yl=0(iii)
If3≤k≤5, then uiui′=ui′ui=p2r0+pui+vj, uivj=vjui=p2r0+pui+pvj+wk, uiwk=wkui=p2r0+pui+pwk+yl, uiyl=ylui=p2r0+pui, vjvj′=vj′vj=p2r0+pvj+yl, vjwk=wkvj=p2r0+pvj+pwk, vjyl=ylvj=p2r0+pvj, wkwk′=wk′wk=p2r0+pwk, wkyl=ylwk=p2r0+pwk, ylyl′=yl′yl=p2r0Furtheruiui′ui′′ui′′′uiiv=0, uir0=r0ui, vjr0=r0vj, wkr0=r0wk, ylr0=r0yl, where r0∈R0 and 1≤i,i′≤e, 1≤j,j′≤f, 1≤k,k′≤g, 1≤l,l′≤h. From the given multiplication in R, we see that if r0+∑i=1eriui+∑j=1fsjvj+∑k=1gtkwk+∑l=1hzlyl and r0′+∑i=1eri′ui+∑j=1fsj′vj+∑k=1gtk′wk+∑l=1hzl′yl are any two elements of R, then
(1)r0+∑i=1eriui+∑j=1fsjvj+∑k=1gtkwk+∑l=1hzlylr0′+∑i=1eri′ui+∑j=1fsj′vj+∑k=1gtk′wk+∑l=1hzl′yl=r0r0′+pa∑i,m=1erirm′+pR0+∑i=1er0ri′+rir0′+pR0ui+∑j=1fr0+pR0sj′+sjr0′+pR0+∑ν,μ=1erνrμ′+pR0vj+∑k=1gr0+pR0tk′+tkr0′+pR0+∑i,jri+pR0sj′+sjri′+pR0wk+∑l=1hr0+pR0zl′+zlr0′+pR0+∑i,kri+pR0tk′+tkri′+pR0+∑κ,τ=1fsκsτ′+pR0yl,where a=1,2,3, or 4 depending on whether Char R0=p2,p3,p4, or p5. It can be verified that this multiplication turns R into a commutative ring with identity 1.Notice that ifR0=GRpr,p where Char R=p, then the above multiplication reduces to
(2)r0+∑i=1eriui+∑j=1fsjvj+∑k=1gtkwk+∑l=1hzlylr0′+∑i=1eri′ui+∑j=1fsj′vj+∑k=1gtk′wk+∑l=1hzl′yl=r0r0′+∑i=1er0ri′+rir0′ui+∑j=1fr0sj′+sjr0′+∑ν,μ=1erνrμ′vj+∑k=1gr0tk′+tkr0′+∑i,jrisj′+sjri′wk+∑l=1hr0zl′+zlr0′+∑i,kritk′+tkri′+∑κ,τ=1fsκsτ′yl.Since the ring(3)R=R0⊕∑i=1eR0ui⊕∑j=1fR0vj⊕∑k=1gR0wk⊕∑l=1hR0ylasR0‐module,we have its maximal ideal as
(4)ZR=pR0⊕∑i=1eR0ui⊕∑j=1fR0vj⊕∑k=1gR0wk⊕∑l=1hR0yl,and the finite abelian p-group given by
(5)1+ZR=1+pR0⊕∑i=1eR0ui⊕∑j=1fR0vj⊕∑k=1gR0wk⊕∑l=1hR0yl.The unit groups ofR is given by R∗, and it can be verified that
(6)R∗=R∗/1+ZR×1+ZR=<b>×1+ZR,where <b> is a cyclic group of order pr−1. The structure problem of R∗ is reduced to that of 1+ZR. Using the ideas of Raghavendran [11] and Chikunji [1], we classify the unit groups of the rings constructed in this section.Proposition 1.
LetR be a ring constructed above and of characteristic p with pui=0, pvj=0, pwk=0, and pyl=0. Then, its group of units is characterized as follows:
(7)R∗≅ℤ2r−1×ℤ8re×ℤ2rg,ifp=2,ℤ3r−1×ℤ9re×ℤ3rf×ℤ3rh,ifp=3,ℤpr−1×ℤpre×ℤprf×ℤprg×ℤprh,ifp>3.Proof.
See Proposition 3.1 in [5].


## 2.1. Construction

LetR0=GRpkr,pk be a Galois ring of order pkr and characteristic pk where p is a prime integer, 1≤k≤5 and r∈ℤ+. Suppose U,V,W, and Y are R0/pR0-spaces considered R0 modules generated by e,f,g, and h elements, respectively, such that the corresponding generating sets are u1,⋯,ue, v1,⋯,vf, w1,⋯,wg, and y1,⋯,yh, so that R=R0⊕U⊕V⊕W⊕Y is an additive abelian group. Then, on the additive group, we define multiplication by the following relations:
(i)
Ifk=1, then uiui′=ui′ui=vj, uivj=vjui=wk, uiwk=wkui=yl, uiyl=ylui=0, vjvj′=vj′vj=yl, vjwk=wkvj=0, vjyl=ylvj=0, wkwk′=wk′wk=0, wkyl=ylwk=0, ylyl′=yl′yl=0(ii)
Ifk=2, then uiui′=ui′ui=pr0+pui+vj, uivj=vjui=pui+wk, uiwk=wkui=pui+yl, uiyl=ylui=pui, vjvj′=vj′vj=yl, vjwk=wkvj=0, vjyl=ylvj=0, wkwk′=wk′wk=0, wkyl=ylwk=0, ylyl′=yl′yl=0(iii)
If3≤k≤5, then uiui′=ui′ui=p2r0+pui+vj, uivj=vjui=p2r0+pui+pvj+wk, uiwk=wkui=p2r0+pui+pwk+yl, uiyl=ylui=p2r0+pui, vjvj′=vj′vj=p2r0+pvj+yl, vjwk=wkvj=p2r0+pvj+pwk, vjyl=ylvj=p2r0+pvj, wkwk′=wk′wk=p2r0+pwk, wkyl=ylwk=p2r0+pwk, ylyl′=yl′yl=p2r0Furtheruiui′ui′′ui′′′uiiv=0, uir0=r0ui, vjr0=r0vj, wkr0=r0wk, ylr0=r0yl, where r0∈R0 and 1≤i,i′≤e, 1≤j,j′≤f, 1≤k,k′≤g, 1≤l,l′≤h. From the given multiplication in R, we see that if r0+∑i=1eriui+∑j=1fsjvj+∑k=1gtkwk+∑l=1hzlyl and r0′+∑i=1eri′ui+∑j=1fsj′vj+∑k=1gtk′wk+∑l=1hzl′yl are any two elements of R, then
(1)r0+∑i=1eriui+∑j=1fsjvj+∑k=1gtkwk+∑l=1hzlylr0′+∑i=1eri′ui+∑j=1fsj′vj+∑k=1gtk′wk+∑l=1hzl′yl=r0r0′+pa∑i,m=1erirm′+pR0+∑i=1er0ri′+rir0′+pR0ui+∑j=1fr0+pR0sj′+sjr0′+pR0+∑ν,μ=1erνrμ′+pR0vj+∑k=1gr0+pR0tk′+tkr0′+pR0+∑i,jri+pR0sj′+sjri′+pR0wk+∑l=1hr0+pR0zl′+zlr0′+pR0+∑i,kri+pR0tk′+tkri′+pR0+∑κ,τ=1fsκsτ′+pR0yl,where a=1,2,3, or 4 depending on whether Char R0=p2,p3,p4, or p5. It can be verified that this multiplication turns R into a commutative ring with identity 1.Notice that ifR0=GRpr,p where Char R=p, then the above multiplication reduces to
(2)r0+∑i=1eriui+∑j=1fsjvj+∑k=1gtkwk+∑l=1hzlylr0′+∑i=1eri′ui+∑j=1fsj′vj+∑k=1gtk′wk+∑l=1hzl′yl=r0r0′+∑i=1er0ri′+rir0′ui+∑j=1fr0sj′+sjr0′+∑ν,μ=1erνrμ′vj+∑k=1gr0tk′+tkr0′+∑i,jrisj′+sjri′wk+∑l=1hr0zl′+zlr0′+∑i,kritk′+tkri′+∑κ,τ=1fsκsτ′yl.Since the ring(3)R=R0⊕∑i=1eR0ui⊕∑j=1fR0vj⊕∑k=1gR0wk⊕∑l=1hR0ylasR0‐module,we have its maximal ideal as
(4)ZR=pR0⊕∑i=1eR0ui⊕∑j=1fR0vj⊕∑k=1gR0wk⊕∑l=1hR0yl,and the finite abelian p-group given by
(5)1+ZR=1+pR0⊕∑i=1eR0ui⊕∑j=1fR0vj⊕∑k=1gR0wk⊕∑l=1hR0yl.The unit groups ofR is given by R∗, and it can be verified that
(6)R∗=R∗/1+ZR×1+ZR=<b>×1+ZR,where <b> is a cyclic group of order pr−1. The structure problem of R∗ is reduced to that of 1+ZR. Using the ideas of Raghavendran [11] and Chikunji [1], we classify the unit groups of the rings constructed in this section.Proposition 1.
LetR be a ring constructed above and of characteristic p with pui=0, pvj=0, pwk=0, and pyl=0. Then, its group of units is characterized as follows:
(7)R∗≅ℤ2r−1×ℤ8re×ℤ2rg,ifp=2,ℤ3r−1×ℤ9re×ℤ3rf×ℤ3rh,ifp=3,ℤpr−1×ℤpre×ℤprf×ℤprg×ℤprh,ifp>3.Proof.
See Proposition 3.1 in [5].


## 3. Main Results

We now investigate the structures of the unit groups of finite completely primary rings with a maximal idealsZR such that ZR5=0,ZR4≠0 and with the characteristics pk,1≤k≤5, under the restrictions pψui=0, pζvj=0, pwk=0, pyl=0, where ψ=2,3,4; ζ=2,3; 1≤i≤e, 1≤j≤f, 1≤k≤g, and 1≤l≤h.Proposition 2.
LetR be a ring constructed above and of characteristic p2 with p2ui=0, p2vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(8)R∗≅ℤ2r−1×ℤ2r×ℤ8re×ℤ4rf×ℤ2rg,p=2,ℤpr−1×ℤpr×ℤp2re×ℤp2rf×ℤprg×ℤprh,p≠2.Proof.
SinceR is commutative, R∗=<b>⋅1+ZR≅<b>×1+ZR, a direct product of the p-group 1+ZR by the cyclic group <b>. Then it suffices to determine the structure of the subgroup 1+ZR of the unit group R∗. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. We consider the two cases separately.Case 1.
p
is even. For each t=1,⋯,r, 1+2ξt2=1, 1+ξtui8=1, 1+ξtvj4=1, 1+ξtwk2=1. For nonnegative integers αt, δt, φt, and λt with αt≤2, δt≤8, φt≤4, and λt≤2, it is clear that
(9)∏t=1r1+2ξtαt⋅∏i=1e∏t=1r1+ξtuiδt⋅∏j=1f∏t=1r1+ξtvjφt⋅∏k=1g∏t=1r1+ξtwkλt=1will imply that αt=2, δt=8, φt=4, and λt=2, for all t=1,⋯,r.
If we setAt=1+2ξtα:α=1,2;∀t=1,⋯,r; Bt,i=1+ξtuiδ:δ=1,⋯,8;∀t=1,⋯,r, i=1,⋯,e; Ct,j=1+ξtvjφ:φ=1,⋯,4;∀t=1,⋯,r,j=1,⋯,f; and Dt,k=1+ξtwkλ:λ=1,2;∀t=1,⋯,r,k=1,⋯,g; we see that At, Bt,i, Ct,j, and Dt,k are all cyclic subgroups of 1+ZR, and these are of the orders inferred from their definition. Since the intersection of any pair of the cyclic subgroups <1+2ξt>, <1+ξtui>, <1+ξtvj>, and <1+ξtwk> is trivial and that
(10)∏t=1r<1+2ξt>⋅∏i=1e∏t=1r<1+ξtui>⋅∏j=1f∏t=1r<1+ξtvj>⋅∏k=1g∏t=1r<1+ξtwk>coincides with 1+ZR, it follows that
(11)1+ZR=∏t=1r<1+2ξt>×∏i=1e∏t=1r<1+ξtui>×∏j=1f∏t=1r<1+ξtvj>×∏k=1g∏t=1r<1+ξtwk>≅ℤ2r×ℤ8re×ℤ4rf×ℤ2rg.Case 2.
p
is odd. For each t=1,⋯,r, 1+pξtp=1, 1+ξtuip2=1, 1+ξtvjp2=1, 1+ξtwkp=1, and 1+ξtylp=1. For nonnegative integers βt, αt,δt, φt, and λt with βt≤p, αt≤p2, δt≤p2, φt≤p, and λt≤p, it is clear that
(12)∏t=1r1+pξtβt⋅∏i=1e∏t=1r1+ξtuiαt⋅∏j=1f∏t=1r1+ξtvjδt⋅∏k=1g∏t=1r1+ξtwkφt⋅∏l=1h∏t=1r1+ξtylλt=1will imply that βt=p, αt=p2, δt=p2, φt=p, and λt=p, for all t=1,⋯,r.
If we setAt=1+pξtβ:β=1,⋯,p;∀t=1,⋯,r; Bt,i=1+ξtuiα:α=1,⋯,p2;∀t=1,⋯,ri=1,⋯,e; Ct,j=1+ξtvjδ:δ=1,⋯,p2;∀t=1,⋯,r,j=1,⋯,f; Dt,k=1+ξtwkφ:φ=1,⋯,p;∀t=1,⋯,r,k=1,⋯,g; and Et,l=1+ξtylλ:λ=1,⋯,p;∀t=1,⋯,r,l=1,⋯,h, we see that At, Bt,i, Ct,j, Dt,k, and Et,l are all cyclic subgroups of 1+ZR, and these are of the orders inferred from their definition. Since the intersection of any pair of the cyclic subgroups <1+pξt>, <1+ξtui>, <1+ξtvj>, <1+ξtwk>, and <1+ξtyl> is trivial and that
(13)∏t=1r<1+pξt>⋅∏i=1e∏t=1r<1+ξtui>⋅∏j=1f∏t=1r<1+ξtvj>⋅∏k=1g∏t=1r<1+ξtwk>⋅∏l=1h∏t=1r<1+ξtyl>coincides with 1+ZR, it follows that
(14)1+ZR=∏t=1r<1+pξt>×∏i=1e∏t=1r<1+ξtui>×∏j=1f∏t=1r<1+ξtvj>×∏k=1g∏t=1r<1+ξtwk>×∏l=1h∏t=1r<1+ξtyl>≅ℤpr×ℤp2re×ℤp2rf×ℤprg×ℤprh.Proposition 3.
LetR be a ring constructed above and of characteristic p3 with p2ui=0, p2vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(15)R∗≅ℤ2r−1×ℤ2r×ℤ2r×ℤ2re×ℤ8re+f×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp2r×ℤp2re×ℤp2rf×ℤprg×ℤprh,p≠2.Proof.
SinceR is commutative, R∗=<b>⋅1+ZR≅<b>×1+ZR, a direct product of the p-group 1+ZR by the cyclic group <b>. Then it suffices to determine the structure of the subgroup 1+ZR of the unit group R∗. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. We consider the two cases separately.Case 1.
p
is even. For each t=1,⋯,r, 1+2ξt2=1, 1+4ξt2=1, 1+2ξtui2=1, 1+ξtui+ξtvj8=1, 1+ξtwk2=1, and 1+ξtyl2=1. For nonnegative integers ηt, βt, αt, δt, φt, and λt with ηt≤2, βt≤2, αt≤2, δt≤8, φt≤2, and λt≤2, it is clear that
(16)∏t=1r1+2ξtηt⋅∏t=1r1+4ξtβt⋅∏i=1e∏t=1r1+2ξtuiαt⋅∏j=1f∏i=1e∏t=1r1+ξtui+ξtvjδt⋅∏k=1g∏t=1r1+ξtwkφt⋅∏l=1h∏t=1r1+ξtylλt=1will imply that ηt=2, βt=2, αt=2, δt=8, φt=2, and λt=2, for all t=1,⋯,r.
If we setAt=1+2ξtη:η=1,2;∀t=1,⋯,r; Bt=1+4ξtβ:β=1,2;∀t=1,⋯,r; Ct,i=1+2ξtuiα:α=1,2;∀t=1,⋯,r,i=1,⋯,e; Dt,i,j=1+ξtui+ξtvjδ:δ=1,⋯,8;∀t=1,⋯,r, i=1,⋯,e,j=1,⋯,f; Et,k=1+ξtwkφ:φ=1,2;∀t=1,⋯,r,k=1,⋯,g; Ft,l=1+ξtylλ:λ=1,2;∀t=1,⋯,r,l=1,⋯,h; we see that At, Bt, Ct,i, Dt,i,j, Et,k, and Ft,l are all cyclic subgroups of 1+ZR, and these are of the orders inferred from their definition. Since the intersection of any pair of the cyclic subgroups 1+2ξt, 1+4ξt, 1+2ξtui, 1+ξtui+ξtvj, 1+ξtwk, and 1+ξtyl is trivial and that
(17)∏t=1r<1+2ξt>⋅∏t=1r<1+4ξt>⋅∏i=1e∏t=1r<1+2ξtui>⋅∏j=1f∏i=1e∏t=1r<1+ξtui+ξtvj>⋅∏k=1g∏t=1r<1+ξtwk>⋅∏l=1h∏t=1r<1+ξtyl>coincides with 1+ZR, it follows that
(18)1+ZR=∏t=1r<1+2ξt>×∏t=1r<1+4ξt>×∏i=1e∏t=1r<1+2ξtui>×∏j=1f∏i=1e∏t=1r<1+ξtui+ξtvj>×∏k=1g∏t=1r<1+ξtwk>×∏l=1h∏t=1r<1+ξtyl>≅ℤ2r×ℤ2r×ℤ2re×ℤ8re+f×ℤ2rg×ℤ2rh.Case 2.
p
is odd. For each t=1,⋯,r, 1+pξtp2=1, 1+ξtuip2=1, 1+ξtvjp2=1, 1+ξtwkp=1, and 1+ξtylp=1. For nonnegative integers βt, αt, δt, φt, and λt with βt≤p2, αt≤p2, δt≤p2, φt≤p, and λt≤p, it is clear that
(19)∏t=1r1+pξtβt⋅∏i=1e∏t=1r1+ξtuiαt⋅∏j=1f∏t=1r1+ξtvjδt⋅∏k=1g∏t=1r1+ξtwkφt⋅∏l=1h∏t=1r1+ξtylλt=1will imply that βt=p2, αt=p2, δt=p2, φt=p, and λt=p, for all t=1,⋯,r.
If we setAt=1+pξtβ:β=1,⋯,p2;∀t=1,⋯,r; Bt,i=1+ξtuiα:α=1,⋯,p2;∀t=1,⋯,r, i=1,⋯,e; Ct,j=1+ξtvjδ:δ=1,⋯,p2;∀t=1,⋯,r,j=1,⋯,f; Dt,k=1+ξtwkφ:φ=1,⋯,p;∀t=1,⋯,r,k=1,⋯,g; and Et,l=1+ξtylλ:λ=1,⋯,p;∀t=1,⋯,r,l=1,⋯,h, we see that At, Bt,i, Ct,jDt,k, and Et,l are all cyclic subgroups of 1+ZR, and these are of the orders inferred from their definition. Since the intersection of any pair of the cyclic subgroups 1+pξt, 1+ξtui, 1+ξtvj, 1+ξtwk, and 1+ξtyl is trivial and that
(20)∏t=1r<1+pξt>⋅∏i=1e∏t=1r<1+ξtui>⋅∏j=1f∏t=1r<1+ξtvj>⋅∏k=1g∏t=1r<1+ξtwk>⋅∏l=1h∏t=1r<1+ξtyl>coincides with 1+ZR, it follows that
(21)1+ZR=∏t=1r<1+pξt>×∏i=1e∏t=1r<1+ξtui>×∏j=1f∏t=1r<1+ξtvj>×∏k=1g∏t=1r<1+ξtwk>×∏l=1h∏t=1r<1+ξtyl>≅ℤp2r×ℤp2re×ℤp2rf×ℤprg×ℤprh.Proposition 4.
LetR be a ring constructed above and of characteristic p3 with p3ui=0, p2vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(22)R∗≅ℤ2r−1×ℤ2r×ℤ2r×ℤ8re×ℤ4rf×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp2r×ℤp3re×ℤp2rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 2; 1+4ξt of order 2; 1+ξtui of order 8; 1+ξtvj of order 4; 1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 3.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p2; 1+ξtui of order p3; 1+ξtvj of order p2; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 3.Proposition 5.
LetR be a ring constructed above and of characteristic p3 with p3ui=0, p3vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(23)R∗≅ℤ2r−1×ℤ2r×ℤ2r×ℤ4re×ℤ16re+f×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp2r×ℤp3re×ℤp3rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 2; 1+4ξt of order 2; 1+2ξtui of order 4; 1+ξtui+ξtvj of order 16;1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 3.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p2; 1+ξtui of order p3; 1+ξtvj of order p3; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 3.Proposition 6.
LetR be a ring constructed above and of characteristic p4 with p2ui=0, p2vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(24)R∗≅ℤ2r−1×ℤ4r×ℤ2r×ℤ2re×ℤ8re+f×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp3r×ℤp2re×ℤp2rf×ℤprg×ℤprh,p≠2.Proof.
SinceR is commutative, R∗=<b>⋅1+ZR≅<b>×1+ZR, a direct product of the p-group 1+ZR by the cyclic group <b>. Then it suffices to determine the structure of the subgroup 1+ZR of the unit group R∗. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. We consider the two cases separately.Case 1.
p
is even. For each t=1,⋯,r, 1+2ξt4=1, 1+6ξt2=1, 1+2ξtui2=1, 1+ξtui+ξtvj8=1, 1+ξtwk2=1, and 1+ξtyl2=1. For nonnegative integers ηt, βt, αt, δt, φt, and λt with ηt≤4, βt≤2, αt≤2, δt≤8, φt≤2, and λt≤2, it is clear that
(25)∏t=1r1+2ξtηt⋅∏t=1r1+6ξtβt⋅∏i=1e∏t=1r1+2ξtuiαt⋅∏j=1f∏i=1e∏t=1r1+ξtui+ξtvjδt⋅∏k=1g∏t=1r1+ξtwkφt⋅∏l=1h∏t=1r1+ξtylλt=1will imply that ηt=4, βt=2, αt=2, δt=8, φt=2, and λt=2, for all t=1,⋯,r.
If we setAt=1+2ξtη:η=1,⋯,4;∀t=1,⋯,r; Bt=1+6ξtβ:β=1,2;∀t=1,⋯,r; Ct,i=1+2ξtuiα:α=1,2;∀t=1,⋯,r,i=1,⋯,e; Dt,i,j=1+ξtui+ξtvjδ:δ=1,⋯,8;∀t=1,⋯,r,i=1,⋯,e,j=1,⋯,f; Et,k=1+ξtwkφ:φ=1,2;∀t=1,⋯,r,k=1,⋯,g; and Ft,l=1+ξtylλ:λ=1,2;∀t=1,⋯,r,l=1,⋯,h, we see that At, Bt, Ct,i, Dt,i,j, Et,k, and Ft,l are all cyclic subgroups of 1+ZR, and these are of the orders inferred from their definition. Since the intersection of any pair of the cyclic subgroups <1+2ξt>, <1+6ξt>, <1+2ξtui>, <1+ξtui+ξtvj>, <1+ξtwk>, and <1+ξtyl> is trivial and that
(26)∏t=1r<1+2ξt>⋅∏t=1r<1+6ξt>⋅∏i=1e∏t=1r<1+2ξtui>⋅∏j=1f∏i=1e∏t=1r<1+ξtui+ξtvj>⋅∏k=1g∏t=1r<1+ξtwk>⋅∏l=1h∏t=1r<1+ξtyl>coincides with 1+ZR, it follows that
(27)1+ZR=∏t=1r<1+2ξt>×∏t=1r<1+6ξt>×∏i=1e∏t=1r<1+2ξtui>×∏j=1f∏i=1e∏t=1r<1+ξtui+ξtvj>×∏k=1g∏t=1r<1+ξtwk>×∏l=1h∏t=1r<1+ξtyl>≅ℤ4r×ℤ2r×ℤ2re×ℤ8re+f×ℤ2rg×ℤ2rh.Case 2.
p
is odd. For each t=1,⋯,r, 1+pξtp3=1, 1+ξtuip2=1, 1+ξtvjp2=1, 1+ξtwkp=1, and 1+ξtylp=1. For nonnegative integers βt, αt, δt, φt, and λt with βt≤p3, αt≤p2, δt≤p2, φt≤p, and λt≤p, it is clear that
(28)∏t=1r1+pξtβt⋅∏i=1e∏t=1r1+ξtuiαt⋅∏j=1f∏t=1r1+ξtvjδt⋅∏k=1g∏t=1r1+ξtwkφt⋅∏l=1h∏t=1r1+ξtylλt=1will imply that βt=p3, αt=p2, δt=p2, φt=p, and λt=p, for all t=1,⋯,r.
If we setAt=1+pξtβ:β=1,⋯,p3;∀t=1,⋯,r; Bt,i=1+ξtuiα:α=1,⋯,p2;∀t=1,⋯,r, i=1,⋯,e; Ct,j=1+ξtvjδ:δ=1,⋯,p2;∀t=1,⋯,r,j=1,⋯,f; Dt,k=1+ξtwkφ:φ=1,⋯,p;∀t=1,⋯,r,k=1,⋯,g; and Et,l=1+ξtylλ:λ=1,⋯,p;∀t=1,⋯,r,l=1,⋯,h, we see that At, Bt,i, Ct,j, Dt,k, and Et,l are all cyclic subgroups of 1+ZR, and these are of the orders inferred from their definition. Since the intersection of any pair of the cyclic subgroups 1+pξt, 1+ξtui, 1+ξtvj, 1+ξtwk, and 1+ξtyl is trivial and that
(29)∏t=1r<1+pξt>⋅∏i=1e∏t=1r<1+ξtui>⋅∏j=1f∏t=1r<1+ξtvj>⋅∏k=1g∏t=1r<1+ξtwk>⋅∏l=1h∏t=1r<1+ξtyl>coincides with 1+ZR, it follows that
(30)1+ZR=∏t=1r<1+pξt>×∏i=1e∏t=1r<1+ξtui>×∏j=1f∏t=1r<1+ξtvj>×∏k=1g∏t=1r<1+ξtwk>×∏l=1h∏t=1r<1+ξtyl>≅ℤp3r×ℤp2re×ℤp2rf×ℤprg×ℤprh.Proposition 7.
LetR be a ring constructed above and of characteristic p4 with p3ui=0, p2vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(31)R∗≅ℤ2r−1×ℤ4r×ℤ2r×ℤ8re×ℤ4rf×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp3r×ℤp3re×ℤp2rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 4; 1+14ξt of order 2; 1+ξtui of order 8; 1+ξtvj of order 4; 1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 6.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p3; 1+ξtui of order p3; 1+ξtvj of order p2; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 6.Proposition 8.
LetR be a ring constructed above and of characteristic p4 with p3ui=0, p3vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(32)R∗≅ℤ2r−1×ℤ4r×ℤ2r×ℤ4re×ℤ16re+f×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp3r×ℤp3re×ℤp3rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 4; 1+6ξt of order 2; 1+2ξtui of order 4; 1+ξtui+ξtvj of order 16;1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 6.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p3; 1+ξtui of order p3; 1+ξtvj of order p3; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 6.Proposition 9.
LetR be a ring constructed above and of characteristic p4 with p4ui=0, p2vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(33)R∗≅ℤ2r−1×ℤ4r×ℤ2r×ℤ16re×ℤ4rf×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp3r×ℤp4re×ℤp2rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 4; 1+6ξt of order 2; 1+ξtui of order 16; 1+ξtvj of order 4; 1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 6.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p3; 1+ξtui of order p4; 1+ξtvj of order p2; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 6.Proposition 10.
LetR be a ring constructed above and of characteristic p4 with p4ui=0, p3vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(34)R∗≅ℤ2r−1×ℤ4r×ℤ2r×ℤ16re×ℤ8rf×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp3r×ℤp4re×ℤp3rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 4; 1+6ξt of order 2; 1+ξtui of order 16; 1+ξtvj of order 8; 1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 6.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p3; 1+ξtui of order p4; 1+ξtvj of order p3; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 6.Proposition 11.
LetR be a ring constructed above and of characteristic p5 with p2ui=0, p2vj=0, pwk=0, andpyl=0. Then its group of units is characterized as follows:
(35)R∗≅ℤ2r−1×ℤ8r×ℤ2r×ℤ2re×ℤ8re+f×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp4r×ℤp2re×ℤp2rf×ℤprg×ℤprh,p≠2.Proof.
SinceR is commutative, R∗=<b>⋅1+ZR≅<b>×1+ZR, a direct product of the p-group 1+ZR by the cyclic group <b>. Then it suffices to determine the structure of the subgroup 1+ZR of the unit group R∗. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. We consider the two cases separately.Case 1.
p
is even. For each t=1,⋯,r, 1+2ξt8=1, 1+14ξt2=1, 1+2ξtui2=1, 1+ξtui+ξtvj8=1, 1+ξtwk2=1, and 1+ξtyl2=1. For nonnegative integers ηt, βt, αt, δt, φt, and λt with ηt≤8, βt≤2, αt≤2, δt≤8, φt≤2, and λt≤2, it is clear that
(36)∏t=1r1+2ξtηt⋅∏t=1r1+14ξtβt⋅∏i=1e∏t=1r1+2ξtuiαt⋅∏j=1f∏i=1e∏t=1r1+ξtui+ξtvjδt⋅∏k=1g∏t=1r1+ξtwkφt⋅∏l=1h∏t=1r1+ξtylλt=1will imply that ηt=8, βt=2, αt=2, δt=8, φt=2, and λt=2 for all t=1,⋯,r.
If we setAt=1+2ξtη:η=1,⋯,8;∀t=1,⋯,r; Bt=1+14ξtβ:β=1,2;∀t=1,⋯,r; Ct,i=1+2ξtuiα:α=1,2;∀t=1,⋯,r,i=1,⋯,e; Dt,i,j=1+ξtui+ξtvjδ:δ=1,⋯,8;∀t=1,⋯,r,i=1,⋯,e,j=1,⋯,f; Et,k=1+ξtwkφ:φ=1,2;∀t=1,⋯,r,k=1,⋯,g; and Ft,l=1+ξtylλ:λ=1,2;∀t=1,⋯,r,l=1,⋯,h, we see that At, Bt, Ct,i, Dt,i,j, Et,k, and Ft,l are all cyclic subgroups of 1+ZR, and these are of the orders inferred from their definition. Since the intersection of any pair of the cyclic subgroups <1+2ξt>, <1+14ξt>, <1+2ξtui>, <1+ξtui+ξtvj>, <1+ξtwk>, and <1+ξtyl> is trivial and that
(37)∏t=1r<1+2ξt>⋅∏t=1r<1+14ξt>⋅∏i=1e∏t=1r<1+2ξtui>⋅∏j=1f∏i=1e∏t=1r<1+ξtui+ξtvj>⋅∏k=1g∏t=1r<1+ξtwk>⋅∏l=1h∏t=1r<1+ξtyl>coincides with 1+ZR, it follows that
(38)1+ZR=∏t=1r<1+2ξt>×∏t=1r<1+14ξt>×∏i=1e∏t=1r<1+2ξtui>×∏j=1f∏i=1e∏t=1r<1+ξtui+ξtvj>×∏k=1g∏t=1r<1+ξtwk>×∏l=1h∏t=1r<1+ξtyl>≅ℤ8r×ℤ2r×ℤ2re×ℤ8re+f×ℤ2rg×ℤ2rh.Case 2.
p
is odd. For each t=1,⋯,r, 1+pξtp4=1, 1+ξtuip2=1, 1+ξtvjp2=1, 1+ξtwkp=1, and 1+ξtylp=1. For nonnegative integers βt, αt, δt, φt, and λt with βt≤p4, αt≤p2, δt≤p2, φt≤p, and λt≤p, it is clear that
(39)∏t=1r1+pξtβt⋅∏i=1e∏t=1r1+ξtuiαt⋅∏j=1f∏t=1r1+ξtvjδt⋅∏k=1g∏t=1r1+ξtwkφt⋅∏l=1h∏t=1r1+ξtylλt=1will imply that βt=p4, αt=p2, δt=p2, φt=p, and λt=p, for all t=1,⋯,r.
If we setAt=1+pξtβ:β=1,⋯,p4;∀t=1,⋯,r; Bt,i=1+ξtuiα:α=1,⋯,p2;∀t=1,⋯,r, i=1,⋯,e; Ct,j=1+ξtvjδ:δ=1,⋯,p2;∀t=1,⋯,r,j=1,⋯,f; Dt,k=1+ξtwkφ:φ=1,⋯,p;∀t=1,⋯,r,k=1,⋯,g; and Et,l=1+ξtylλ:λ=1,⋯,p;∀t=1,⋯,r,l=1,⋯,h, we see that At, Bt,i, Ct,j, Dt,k, and Et,l are all cyclic subgroups of 1+ZR, and these are of the orders inferred from their definition. Since the intersection of any pair of the cyclic subgroups <1+pξt>, <1+ξtui>, <1+ξtvj>, <1+ξtwk>, and <1+ξtyl> is trivial and that
(40)∏t=1r<1+pξt>⋅∏i=1e∏t=1r<1+ξtui>⋅∏j=1f∏t=1r<1+ξtvj>⋅∏k=1g∏t=1r<1+ξtwk>⋅∏l=1h∏t=1r<1+ξtyl>coincides with 1+ZR, it follows that
(41)1+ZR=∏t=1r<1+pξt>×∏i=1e∏t=1r<1+ξtui>×∏j=1f∏t=1r<1+ξtvj>×∏k=1g∏t=1r<1+ξtwk>×∏l=1h∏t=1r<1+ξtyl>≅ℤp4r×ℤp2re×ℤp2rf×ℤprg×ℤprh.Proposition 12.
LetR be a ring constructed above and of characteristic p5 with p3ui=0, p2vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(42)R∗≅ℤ2r−1×ℤ8r×ℤ2r×ℤ8re×ℤ4rf×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp4r×ℤp3re×ℤp2rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 8; 1+14ξt of order 2; 1+ξtui of order 8; 1+ξtvj of order 4; 1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 11.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p4; 1+ξtui of order p3; 1+ξtvj of order p2; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 11.Proposition 13.
LetR be a ring constructed above and of characteristic p5 with p3ui=0, p3vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(43)R∗≅ℤ2r−1×ℤ8r×ℤ2r×ℤ4re×ℤ16re+f×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp4r×ℤp3re×ℤp3rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 8; 1+14ξt of order 2; 1+2ξtui of order 4; 1+ξtui+ξtvj of order 16; 1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 11.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p4; 1+ξtui of order p3; 1+ξtvj of order p3; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 3.Proposition 14.
LetR be a ring constructed above and of characteristic p5 with p4ui=0, p2vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(44)R∗≅ℤ2r−1×ℤ8r×ℤ2r×ℤ16re×ℤ4rf×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp4r×ℤp4re×ℤp2rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 8; 1+14ξt of order 2; 1+ξtui of order 16; 1+ξtvj of order 4; 1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 11.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p4; 1+ξtui of order p4; 1+ξtvj of order p2; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 11.Proposition 15.
LetR be a ring constructed above and of characteristic p5 withp4ui=0, p3vj=0, pwk=0, and pyl=0. Then its group of units is characterized as follows:
(45)R∗≅ℤ2r−1×ℤ8r×ℤ2r×ℤ8re×ℤ16re+f×ℤ2rg×ℤ2rh,p=2,ℤpr−1×ℤp4r×ℤp4re×ℤp3rf×ℤprg×ℤprh,p≠2.Proof.
SinceR∗≅ℤpr−1×1+ZR, it suffices to determine the structure of 1+ZR. Let ξ1,⋯,ξr be elements of R0 with ξ1=1 so that ξ¯1,⋯,ξ¯r∈R0/pR0≅GFpr form a basis for GFpr regarded as a vector space over its prime subfield GFp. Then the generators with their respective orders are as indicated below.Case 1.
p
is even; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+2ξt of order 8; 1+14ξt of order 2; 1+2ξtui of order 8; 1+ξtui+ξtvj of order 16; 1+ξtwk of order 2; and 1+ξtyl of order 2. The rest of the proof follows a similar argument and may be deduced from Proposition 11.Case 2.
p
is odd; 1≤t≤r, 1≤i≤e, 1≤j≤f, 1≤k≤g, 1≤l≤h, the generators are 1+pξt of order p4; 1+ξtui of order p4; 1+ξtvj of order p3; 1+ξtwk of order p; and 1+ξtyl of order p. The rest of the proof follows a similar argument and may be deduced from Proposition 11.


## 4. Conclusions

The unit groups of some classes of five radical zero commutative completely primary finite rings whose first and second Galois ring module generators are of orderpk,k=2,3,4, have been classified in this work. It is evident that the results are in piece when p=2 and p≥3. Since the unit groups of classes of five radical zero commutative completely primary finite rings in this work have been classified via fundamental theorem of finitely generated abelian group, the use of j-diagram technique by References [8–10] is therefore recommended for possible further study of the unit groups of such rings.



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*Source: 7867431-2022-09-19.xml*