Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring
A finite ring with an identity whose lattice of ideals forms a unique chain is called a finite chain ring. Let <i>R</i> be a commutative chain ring with invariants <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semant...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/2/307 |
| _version_ | 1797396849613602816 |
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| author | Sami Alabiad Yousef Alkhamees |
| author_facet | Sami Alabiad Yousef Alkhamees |
| author_sort | Sami Alabiad |
| collection | DOAJ |
| description | A finite ring with an identity whose lattice of ideals forms a unique chain is called a finite chain ring. Let <i>R</i> be a commutative chain ring with 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>r</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi><mo>.</mo></mrow></semantics></math></inline-formula> It is known that <i>R</i> is an Eisenstein extension of degree <i>k</i> of a Galois ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>=</mo><mi>G</mi><mi>R</mi><mo>(</mo><msup><mi>p</mi><mi>n</mi></msup><mo>,</mo><mi>r</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> does not divide <i>k</i>, the structure of the unit group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is known. The case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>∣</mo><mi>k</mi></mrow></semantics></math></inline-formula> was partially considered by M. Luis (1991) by providing counterexamples demonstrated that the results of Ayoub failed to capture the direct decomposition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> In this article, we manage to determine the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>∣</mo><mi>k</mi></mrow></semantics></math></inline-formula> by fixing Ayoub’s approach. We also sharpen our results by introducing a system of generators for the unit group and enumerating the generators of the same order. |
| first_indexed | 2024-03-09T00:57:30Z |
| format | Article |
| id | doaj.art-477fe8209ff04388a90b6c99ec80cf98 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T00:57:30Z |
| publishDate | 2021-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-477fe8209ff04388a90b6c99ec80cf982023-12-11T16:46:58ZengMDPI AGSymmetry2073-89942021-02-0113230710.3390/sym13020307Recapturing the Structure of Group of Units of Any Finite Commutative Chain RingSami Alabiad0Yousef Alkhamees1Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi ArabiaA finite ring with an identity whose lattice of ideals forms a unique chain is called a finite chain ring. Let <i>R</i> be a commutative chain ring with 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>r</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi><mo>.</mo></mrow></semantics></math></inline-formula> It is known that <i>R</i> is an Eisenstein extension of degree <i>k</i> of a Galois ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>=</mo><mi>G</mi><mi>R</mi><mo>(</mo><msup><mi>p</mi><mi>n</mi></msup><mo>,</mo><mi>r</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> does not divide <i>k</i>, the structure of the unit group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is known. The case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>∣</mo><mi>k</mi></mrow></semantics></math></inline-formula> was partially considered by M. Luis (1991) by providing counterexamples demonstrated that the results of Ayoub failed to capture the direct decomposition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> In this article, we manage to determine the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>∣</mo><mi>k</mi></mrow></semantics></math></inline-formula> by fixing Ayoub’s approach. We also sharpen our results by introducing a system of generators for the unit group and enumerating the generators of the same order.https://www.mdpi.com/2073-8994/13/2/307finite chain ringsgroup of unitsGalois ringsj-diagrams |
| spellingShingle | Sami Alabiad Yousef Alkhamees Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring Symmetry finite chain rings group of units Galois rings j-diagrams |
| title | Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring |
| title_full | Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring |
| title_fullStr | Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring |
| title_full_unstemmed | Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring |
| title_short | Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring |
| title_sort | recapturing the structure of group of units of any finite commutative chain ring |
| topic | finite chain rings group of units Galois rings j-diagrams |
| url | https://www.mdpi.com/2073-8994/13/2/307 |
| work_keys_str_mv | AT samialabiad recapturingthestructureofgroupofunitsofanyfinitecommutativechainring AT yousefalkhamees recapturingthestructureofgroupofunitsofanyfinitecommutativechainring |