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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
Abstract

Finite rings of order pn where n 5 have previously been completely classified up to isomorphism. Given a finite local ring R of order pn and characteristic pk with Jacobson radical J, RJ𝔽pr for some r. For n = 6, r {1,2,3,6}. In this paper, local rings of order p6 with r {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