Products of commutators in matrix rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913333537406976 |
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| author | Brešar, Matej Gardella, Eusebio Thiel, Hannes |
| author_facet | Brešar, Matej Gardella, Eusebio Thiel, Hannes |
| contents | Let $R$ be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when $R$ is commutative, is provided. If, however, $R$ has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if $R$ is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If $R$ is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of $a$ is greater than $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Products of commutators in matrix rings Brešar, Matej Gardella, Eusebio Thiel, Hannes Rings and Algebras Operator Algebras Primary 16S50. Secondary 15A30, 16K40, 16W25, 46L05, 46L10 Let $R$ be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when $R$ is commutative, is provided. If, however, $R$ has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if $R$ is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If $R$ is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of $a$ is greater than $2$. |
| title | Products of commutators in matrix rings |
| topic | Rings and Algebras Operator Algebras Primary 16S50. Secondary 15A30, 16K40, 16W25, 46L05, 46L10 |
| url | https://arxiv.org/abs/2404.18116 |