Classification of Irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Modules of a <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Ring Using Matrix Equations
This paper aims to investigate and categorize all inequivalent and irreducible <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/12/2598 |
| _version_ | 1797455197771923456 |
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| author | Zhichao Chen Ruju Zhao |
| author_facet | Zhichao Chen Ruju Zhao |
| author_sort | Zhichao Chen |
| collection | DOAJ |
| description | This paper aims to investigate and categorize all inequivalent and irreducible <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-modules of a commutative unit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>, equipped with set {1, <i>x</i>, <i>y</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>y</mi></mrow></semantics></math></inline-formula>} satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> as a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-basis by using matrix equations, which was part of a call for a Special Issue about matrix inequalities and equations by Symmetry. If the rank of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-module <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula>, we prove that there are finitely many inequivalent and irreducible <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-modules, respectively, one and three. However, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>, there is no irreducible <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-module. |
| first_indexed | 2024-03-09T15:48:04Z |
| format | Article |
| id | doaj.art-cf5f1dff6ecd44c881830167dd77d737 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T15:48:04Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-cf5f1dff6ecd44c881830167dd77d7372023-11-24T18:19:47ZengMDPI AGSymmetry2073-89942022-12-011412259810.3390/sym14122598Classification of Irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Modules of a <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Ring Using Matrix EquationsZhichao Chen0Ruju Zhao1School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, ChinaCollege of Science, Beibu Gulf University, Qinzhou 535011, ChinaThis paper aims to investigate and categorize all inequivalent and irreducible <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-modules of a commutative unit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>, equipped with set {1, <i>x</i>, <i>y</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>y</mi></mrow></semantics></math></inline-formula>} satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> as a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-basis by using matrix equations, which was part of a call for a Special Issue about matrix inequalities and equations by Symmetry. If the rank of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-module <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula>, we prove that there are finitely many inequivalent and irreducible <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-modules, respectively, one and three. However, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>, there is no irreducible <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></semantics></math></inline-formula>-module.https://www.mdpi.com/2073-8994/14/12/2598<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm997"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>-ringirreducible <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm996"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>-modulematrix equationNIM solution |
| spellingShingle | Zhichao Chen Ruju Zhao Classification of Irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Modules of a <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Ring Using Matrix Equations Symmetry <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm997"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>-ring irreducible <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm996"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>-module matrix equation NIM solution |
| title | Classification of Irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Modules of a <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Ring Using Matrix Equations |
| title_full | Classification of Irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Modules of a <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Ring Using Matrix Equations |
| title_fullStr | Classification of Irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Modules of a <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Ring Using Matrix Equations |
| title_full_unstemmed | Classification of Irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Modules of a <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Ring Using Matrix Equations |
| title_short | Classification of Irreducible <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Modules of a <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>-Ring Using Matrix Equations |
| title_sort | classification of irreducible inline formula math display inline semantics mrow msub mi mathvariant double struck z mi mo mo msub mrow semantics math inline formula modules of a inline formula math display inline semantics mrow msub mi mathvariant double struck z mi mo mo msub mrow semantics math inline formula ring using matrix equations |
| topic | <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm997"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>-ring irreducible <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm996"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content>-module matrix equation NIM solution |
| url | https://www.mdpi.com/2073-8994/14/12/2598 |
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