Structure of a chain ring as a ring of matrices over a Galois ring

Yousef Alkhamees; Badr Alhajouj; Yousef Alkhamees; Badr Alhajouj

doi:10.3934/math.2022866

AIMS Mathematics

2022, Volume 7, Issue 9: 15824-15833. doi: 10.3934/math.2022866

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Research article

Structure of a chain ring as a ring of matrices over a Galois ring

Yousef Alkhamees ^,,
Badr Alhajouj

Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia

Received: 16 February 2022 Revised: 24 May 2022 Accepted: 12 June 2022 Published: 27 June 2022
MSC : 16L30, 16P20, 16P30

The structure of a finite chain ring has already been described by Wirt in 1972 and others later. The purpose of this article is to describe another structure of a finite chain ring as a ring of square matrices over Galois ring using the companion matrix of a certain Eisenstein polynomial over Galois ring. Such a companion matrix generates the unique maximal ideal of the corresponding matrix chain ring.
- local ring,
- chain ring,
- Galois ring,
- companion matrix
Citation: Yousef Alkhamees, Badr Alhajouj. Structure of a chain ring as a ring of matrices over a Galois ring[J]. AIMS Mathematics, 2022, 7(9): 15824-15833. doi: 10.3934/math.2022866

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Abstract

The structure of a finite chain ring has already been described by Wirt in 1972 and others later. The purpose of this article is to describe another structure of a finite chain ring as a ring of square matrices over Galois ring using the companion matrix of a certain Eisenstein polynomial over Galois ring. Such a companion matrix generates the unique maximal ideal of the corresponding matrix chain ring.

References

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