Unit Groups of Classes of Five Radical Zero Commutative Completely Primary Finite Rings

Were, Hezron Saka; Owino, Maurice Oduor; Gichuki, Moses N.

doi:10.9734/jamcs/2021/v36i830396

Cited by 2 publications

(3 citation statements)

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“…In [3], the authors determined the structure of the unit groups of completely primary finite rings in which the product of any four zero divisors is zero. Were et al in [4] obtained the structures of the group of units of a completely primary finite rings in which the product of any five zero divisors is zero satisfying p ξ ui = pvj = pw k = py l = 0, where ξ = 1, 2, 3, 4; 1 ≤ i ≤ e, 1 ≤ j ≤ f, 1 ≤ k ≤ g, and 1 ≤ l ≤ h. Some of the previously studied related work can be obtained from [5,1,6]. Unless otherwise stated, R shall denote a finite ring, Z(R) its Jacobson radical and R * the group of units of R. If a is an element of R * , then < a > denotes the cyclic group generated by a.…”

Section: Introductionmentioning

confidence: 98%

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

Were

Oduor

Gichuki

2022

ARJOM

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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 \(1 \leq l \leq h\).

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Section: Introductionmentioning

confidence: 98%

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

Were

Oduor

Gichuki

2022

ARJOM

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“…The unit groups of completely primary finite rings with maximal ideal ZðRÞ such that ðZðRÞÞ 3 = ð0Þ with ðZðRÞÞ 2 ≠ ð0Þ have been classified by Chikunji [1][2][3]. Oduor and Onyango [4] constructed a class of completely primary finite rings in which ðZðRÞÞ 4 = ð0Þ with ðZðRÞÞ 3 ≠ ð0Þ 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 ðZðRÞÞ 5 = ð0Þ; ðZðRÞÞ 4 ≠ ð0Þ and further determined the unit groups restricted to some conditions. This construction involved idealization of R 0 -modules, whose choice was based on Wilson [6].…”

Section: Introductionmentioning

confidence: 99%

“…We shall also denote the coefficient Galois subring GRðp kr , p k Þ of characteristic p k and order p kr of the ring R by R 0 . For the previous related work, we refer to [2,3,5,10,[11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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

Were

Oduor

2022

Journal of Mathematics

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Let R 0 = G R p k r , p k be a Galois maximal subring of R so that R = R 0 ⊕ U ⊕ V ⊕ W ⊕ Y , where U , V , W , and Y are R 0 / p R 0 spaces considered as R 0 -modules, generated by the sets u 1 , ⋯ , u e , v 1 , ⋯ , v f , w 1 , ⋯ , w g , and y 1 , ⋯ , y h , respectively. Then, R is a completely primary finite ring with a Jacobson radical Z R such that Z R 5 = 0 and Z R 4 ≠ 0 . The residue field R / Z R is a finite field G F p r for some prime p and positive integer r . The characteristic of R is p k , 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 p ψ u i = 0 , ψ = 2 , 3 , 4 ; p ζ v j = 0 , ζ = 2 , 3 ; p w k = 0 , and p y l = 0 ; 1 ≤ i ≤ e , 1 ≤ j ≤ f , 1 ≤ k ≤ g , and 1 ≤ l ≤ h .

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Unit Groups of Classes of Five Radical Zero Commutative Completely Primary Finite Rings

Cited by 2 publications

References 4 publications

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

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

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

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Unit Groups of Classes of Five Radical Zero Commutative Completely Primary Finite Rings

Cited by 2 publications

References 4 publications

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

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

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

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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