On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $
Let $ M_{n, m}: = Mat_n(\mathbb{Z}/m\mathbb{Z}) $ be the ring of matrices of $ n\times n $ over $ \mathbb{Z}/m\mathbb{Z} $ and $ G_{n, m}: = Gl_n(\mathbb{Z}/m\mathbb{Z}) $ be the multiplicative group of units of $ M_{n, m} $ with $ n\geqslant 2, m\geqslant 2. $ In this paper, we obtain an exact form...
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AIMS Press
2025-03-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2025059 |
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| author | Yifan Luo Kaisheng Lei Qingzhong Ji |
| author_facet | Yifan Luo Kaisheng Lei Qingzhong Ji |
| author_sort | Yifan Luo |
| collection | DOAJ |
| description | Let $ M_{n, m}: = Mat_n(\mathbb{Z}/m\mathbb{Z}) $ be the ring of matrices of $ n\times n $ over $ \mathbb{Z}/m\mathbb{Z} $ and $ G_{n, m}: = Gl_n(\mathbb{Z}/m\mathbb{Z}) $ be the multiplicative group of units of $ M_{n, m} $ with $ n\geqslant 2, m\geqslant 2. $ In this paper, we obtain an exact formula for the number of representations of any element of $ M_{2, m} $ as the sum of $ k $ units in $ M_{2, m} $. Furthermore, by using the technique of Fourier transformation, we also give a formula for the case $ n\ge3 $ and $ m = p $ is a prime. |
| format | Article |
| id | doaj-art-e81a4a90a77d40e2a2727f2a7eaecb1c |
| institution | OA Journals |
| issn | 2688-1594 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj-art-e81a4a90a77d40e2a2727f2a7eaecb1c2025-08-20T02:08:24ZengAIMS PressElectronic Research Archive2688-15942025-03-013331323133210.3934/era.2025059On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $Yifan Luo0Kaisheng Lei1Qingzhong Ji2Department of Mathematics, Nanjing University, Nanjing 210000, ChinaDepartment of Mathematics, Nanjing University, Nanjing 210000, ChinaDepartment of Mathematics, Nanjing University, Nanjing 210000, ChinaLet $ M_{n, m}: = Mat_n(\mathbb{Z}/m\mathbb{Z}) $ be the ring of matrices of $ n\times n $ over $ \mathbb{Z}/m\mathbb{Z} $ and $ G_{n, m}: = Gl_n(\mathbb{Z}/m\mathbb{Z}) $ be the multiplicative group of units of $ M_{n, m} $ with $ n\geqslant 2, m\geqslant 2. $ In this paper, we obtain an exact formula for the number of representations of any element of $ M_{2, m} $ as the sum of $ k $ units in $ M_{2, m} $. Furthermore, by using the technique of Fourier transformation, we also give a formula for the case $ n\ge3 $ and $ m = p $ is a prime.https://www.aimspress.com/article/doi/10.3934/era.2025059rings of matricesfinite fieldssums of unitsfourier transformation |
| spellingShingle | Yifan Luo Kaisheng Lei Qingzhong Ji On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $ Electronic Research Archive rings of matrices finite fields sums of units fourier transformation |
| title | On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $ |
| title_full | On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $ |
| title_fullStr | On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $ |
| title_full_unstemmed | On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $ |
| title_short | On the sumsets of units in a ring of matrices over $ \mathbb{Z}/m\mathbb{Z} $ |
| title_sort | on the sumsets of units in a ring of matrices over mathbb z m mathbb z |
| topic | rings of matrices finite fields sums of units fourier transformation |
| url | https://www.aimspress.com/article/doi/10.3934/era.2025059 |
| work_keys_str_mv | AT yifanluo onthesumsetsofunitsinaringofmatricesovermathbbzmmathbbz AT kaishenglei onthesumsetsofunitsinaringofmatricesovermathbbzmmathbbz AT qingzhongji onthesumsetsofunitsinaringofmatricesovermathbbzmmathbbz |