Classification of chain rings

Yousef Alkhamees; Sami Alabiad; Yousef Alkhamees; Sami Alabiad

doi:10.3934/math.2022284

AIMS Mathematics

2022, Volume 7, Issue 4: 5106-5116. doi: 10.3934/math.2022284

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Research article

Classification of chain rings

Yousef Alkhamees ,
Sami Alabiad ^,

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

Received: 21 October 2021 Revised: 09 December 2021 Accepted: 13 December 2021 Published: 30 December 2021
MSC : 16L30, 16P20, 16P30

An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.
- local ring,
- chain ring,
- Galois ring,
- p-adic field,
- isomorphism class
Citation: Yousef Alkhamees, Sami Alabiad. Classification of chain rings[J]. AIMS Mathematics, 2022, 7(4): 5106-5116. doi: 10.3934/math.2022284

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Abstract

An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.

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