| Title: | Products of commutators in matrix rings |
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| Authors: | ID Brešar, Matej (Author)
ID Gardella, Eusebio (Author)
ID Thiel, Hannes (Author) |
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| Files: |  PDF - Presentation file, download (381,72 KB)
MD5: 7837A45086389740007351A529851B17
 URL - Source URL, visit https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/products-of-commutators-in-matrix-rings/10FD7B61EB100163AA3815437915BA66
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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 $2$. |
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| Keywords: | commutators, matrix ring, division ring, Bass stable rank, L'vov–Kaplansky conjecture |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.06.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | str. 512-529 |
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| Numbering: | Vol. 68, iss. 2 |
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| PID: | 20.500.12556/DiRROS-22101  |
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| UDC: | 512 |
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| ISSN on article: | 0008-4395 |
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| DOI: | 10.4153/S0008439524000523 |
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| COBISS.SI-ID: | 222178051 |
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| Note: |
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| Publication date in DiRROS: | 24.04.2025 |
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| Views: | 479 |
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| Downloads: | 293 |
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