Model-theoretic $K_1$ for modules over semisimple rings: (weak) Morita invariance
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| Format: | Preprint |
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2025
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| _version_ | 1866911253764505600 |
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| author | Banerjee, Sourayan Kuber, Amit |
| author_facet | Banerjee, Sourayan Kuber, Amit |
| contents | This paper is a sequel to a paper by the same authors, where they defined $K$-groups of model-theoretic structures, and computed $K_1$ of free modules over PIDs. In this paper, we compute $K_1$ of a right $M_q(R)$-module $M$, where $R$ is a division ring, $q\geq1$, and $|M_q(R)|\neq 2$. As a consequence, we obtain a (weak) Morita invariance $K_1(R_R)\cong K_1((M_q(R))_{M_q(R)})$ for all division rings $R$ and $q\geq 1$. Finally, we compute $K_1$ of a module over a semisimple ring by showing that the model-theoretic $K_1$ commutes with finite product of modules. We also show that the algebraic $K_1$ of a finite product of infinite matrix rings embeds into the model-theoretic $K_1$ of their right regular modules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_05180 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Model-theoretic $K_1$ for modules over semisimple rings: (weak) Morita invariance Banerjee, Sourayan Kuber, Amit Logic K-Theory and Homology 03C60, 19B99, 19B14 This paper is a sequel to a paper by the same authors, where they defined $K$-groups of model-theoretic structures, and computed $K_1$ of free modules over PIDs. In this paper, we compute $K_1$ of a right $M_q(R)$-module $M$, where $R$ is a division ring, $q\geq1$, and $|M_q(R)|\neq 2$. As a consequence, we obtain a (weak) Morita invariance $K_1(R_R)\cong K_1((M_q(R))_{M_q(R)})$ for all division rings $R$ and $q\geq 1$. Finally, we compute $K_1$ of a module over a semisimple ring by showing that the model-theoretic $K_1$ commutes with finite product of modules. We also show that the algebraic $K_1$ of a finite product of infinite matrix rings embeds into the model-theoretic $K_1$ of their right regular modules. |
| title | Model-theoretic $K_1$ for modules over semisimple rings: (weak) Morita invariance |
| topic | Logic K-Theory and Homology 03C60, 19B99, 19B14 |
| url | https://arxiv.org/abs/2511.05180 |