Commutative Chain Rings with Index of Nilpotency 5 and Residue Field <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><msup><mi>p</mi><mi>m</mi></msup></msub></mrow></semantics></math></inline-formula>
This paper gives a thorough characterization of chain rings with index of nilpotency 5 and residue field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck&q...
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2024-12-01
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| author | Alhanouf Ali Alhomaidhi Sami Alabiad Nawal A. Alsarori |
| author_facet | Alhanouf Ali Alhomaidhi Sami Alabiad Nawal A. Alsarori |
| author_sort | Alhanouf Ali Alhomaidhi |
| collection | DOAJ |
| description | This paper gives a thorough characterization of chain rings with index of nilpotency 5 and 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> represents a prime number, contributing valuable insights to the field of algebraic structures. It carefully identifies and categorizes the family of chain rings with these specifications, thereby enhancing the understanding of their properties and applications. In addition, the work offers a detailed enumeration of all chain rings containing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mrow><mn>5</mn><mi>m</mi></mrow></msup></semantics></math></inline-formula> elements. The significance of finite chain rings is emphasized, particularly in their suitability for coding theory, which confirms their relevance in contemporary mathematical and engineering contexts. |
| format | Article |
| id | doaj-art-24892cb6015e45b2ae379f59b7286d46 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-24892cb6015e45b2ae379f59b7286d462025-08-20T02:01:00ZengMDPI AGAxioms2075-16802024-12-01131287710.3390/axioms13120877Commutative Chain Rings with Index of Nilpotency 5 and Residue Field <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><msup><mi>p</mi><mi>m</mi></msup></msub></mrow></semantics></math></inline-formula>Alhanouf Ali Alhomaidhi0Sami Alabiad1Nawal A. Alsarori2Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, IndiaThis paper gives a thorough characterization of chain rings with index of nilpotency 5 and 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> represents a prime number, contributing valuable insights to the field of algebraic structures. It carefully identifies and categorizes the family of chain rings with these specifications, thereby enhancing the understanding of their properties and applications. In addition, the work offers a detailed enumeration of all chain rings containing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mrow><mn>5</mn><mi>m</mi></mrow></msup></semantics></math></inline-formula> elements. The significance of finite chain rings is emphasized, particularly in their suitability for coding theory, which confirms their relevance in contemporary mathematical and engineering contexts.https://www.mdpi.com/2075-1680/13/12/877chain ringscoding over ringslocal ringsisomorphism classes |
| spellingShingle | Alhanouf Ali Alhomaidhi Sami Alabiad Nawal A. Alsarori Commutative Chain Rings with Index of Nilpotency 5 and Residue Field <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><msup><mi>p</mi><mi>m</mi></msup></msub></mrow></semantics></math></inline-formula> Axioms chain rings coding over rings local rings isomorphism classes |
| title | Commutative Chain Rings with Index of Nilpotency 5 and Residue Field <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><msup><mi>p</mi><mi>m</mi></msup></msub></mrow></semantics></math></inline-formula> |
| title_full | Commutative Chain Rings with Index of Nilpotency 5 and Residue Field <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><msup><mi>p</mi><mi>m</mi></msup></msub></mrow></semantics></math></inline-formula> |
| title_fullStr | Commutative Chain Rings with Index of Nilpotency 5 and Residue Field <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><msup><mi>p</mi><mi>m</mi></msup></msub></mrow></semantics></math></inline-formula> |
| title_full_unstemmed | Commutative Chain Rings with Index of Nilpotency 5 and Residue Field <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><msup><mi>p</mi><mi>m</mi></msup></msub></mrow></semantics></math></inline-formula> |
| title_short | Commutative Chain Rings with Index of Nilpotency 5 and Residue Field <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><msup><mi>p</mi><mi>m</mi></msup></msub></mrow></semantics></math></inline-formula> |
| title_sort | commutative chain rings with index of nilpotency 5 and residue field inline formula math display inline semantics mrow msub mi mathvariant double struck f mi msup mi p mi mi m mi msup msub mrow semantics math inline formula |
| topic | chain rings coding over rings local rings isomorphism classes |
| url | https://www.mdpi.com/2075-1680/13/12/877 |
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