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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2024-12-01
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| author | Sami Alabiad Alhanouf Ali Alhomaidhi Nawal A. Alsarori |
| author_facet | Sami Alabiad Alhanouf Ali Alhomaidhi Nawal A. Alsarori |
| author_sort | Sami Alabiad |
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| description | 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> |
| format | Article |
| id | doaj-art-2faa478e59ea421c8c1e7d8446cd03aa |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-2faa478e59ea421c8c1e7d8446cd03aa2025-01-24T13:22:08ZengMDPI AGAxioms2075-16802024-12-011411210.3390/axioms14010012Finite Local Rings of Length 4Sami Alabiad0Alhanouf Ali Alhomaidhi1Nawal 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 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>https://www.mdpi.com/2075-1680/14/1/12finite ringscoding over ringslocal ringsisomorphism classes |
| spellingShingle | Sami Alabiad Alhanouf Ali Alhomaidhi Nawal A. Alsarori Finite Local Rings of Length 4 Axioms finite rings coding over rings local rings isomorphism classes |
| title | Finite Local Rings of Length 4 |
| title_full | Finite Local Rings of Length 4 |
| title_fullStr | Finite Local Rings of Length 4 |
| title_full_unstemmed | Finite Local Rings of Length 4 |
| title_short | Finite Local Rings of Length 4 |
| title_sort | finite local rings of length 4 |
| topic | finite rings coding over rings local rings isomorphism classes |
| url | https://www.mdpi.com/2075-1680/14/1/12 |
| work_keys_str_mv | AT samialabiad finitelocalringsoflength4 AT alhanoufalialhomaidhi finitelocalringsoflength4 AT nawalaalsarori finitelocalringsoflength4 |