A partial classification of local rings of order p6

Akers, Connor; Szabo, Steve

doi:10.2140/involve.2023.16.151

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A partial classification of local rings of order $p^6$

Connor Akers and Steve Szabo

Vol. 16 (2023), No. 1, 151–165

DOI: 10.2140/involve.2023.16.151

Abstract

Finite rings of order $p^{n}$ where $n \leq 5$ have previously been completely classified up to isomorphism. Given a finite local ring $R$ of order $p^{n}$ and characteristic $p^{k}$ with Jacobson radical $J$ , $R ∕ J ≅ 𝔽_{p^{r}}$ for some $r$ . For $n = 6$ , $r \in {1, 2, 3, 6}$ . In this paper, local rings of order $p^{6}$ with $r \in {2, 3, 6}$ are completely classified. Furthermore, for $r = 1$ , chain rings are classified fully. Many of the remaining cases are also completed. For some of the incomplete cases, techniques for solving them are outlined.

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Keywords

finite rings, local rings, finite ring classification, rings of small order

Mathematical Subject Classification

Primary: 16P10

Milestones

Received: 7 December 2021

Accepted: 20 April 2022

Published: 14 April 2023

Communicated by Nathan Kaplan

Authors

Connor Akers
Department of Mathematics and Statistics Eastern Kentucky University Richmond, KY United States

Steve Szabo
Department of Mathematics and Statistics Eastern Kentucky University Richmond, KY United States

Vol. 16, No. 1, 2023