Finite Local Rings of Length 4

Finite Local Rings of Length 4

This paper presents a comprehensive characterization of finite local rings of length 4 and with residue field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-str...

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Bibliographic Details
Main Authors: Sami Alabiad, Alhanouf Ali Alhomaidhi, Nawal A. Alsarori
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Axioms
Subjects:
finite rings
coding over rings
local rings
isomorphism classes
Online Access:https://www.mdpi.com/2075-1680/14/1/12
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Summary:This paper presents a comprehensive characterization of finite local rings of length 4 and with residue field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><msup><mi>p</mi><mi>m</mi></msup></msub><mo>,</mo></mrow></semantics></math></inline-formula> where <i>p</i> is a prime number. Such rings have an order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mrow><mn>4</mn><mi>m</mi></mrow></msup></semantics></math></inline-formula> elements. The current paper provides the structure and classification, up to isomorphism, of local rings consisting of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mrow><mn>4</mn><mi>m</mi></mrow></msup></semantics></math></inline-formula> elements. We also give the exact number of non-isomorphic classes of these rings with fixed invariants <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>k</mi><mo>.</mo></mrow></semantics></math></inline-formula> In particular, we have listed all finite local rings of 4-length and of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mn>8</mn></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>256</mn><mo>.</mo></mrow></semantics></math></inline-formula>
ISSN:2075-1680

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