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. (19th September 2022)

• Record Type: Journal Article
• Authors: Were, Hezron Saka; Oduor, Maurice Owino
• Other Names: Tartarone Francesca Academic Editor.
• Is Part Of: Journal of mathematics. Volume 2022(2022)
• Publication Date: 2022-09-19
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1155/2022/7867431
• Languages: English
• ISSNs: 2314-4629

Abstract

Abstract : 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 .