Matrix invertible extensions over commutative rings. Part I: general theory
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| Format: | Preprint |
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2024
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| _version_ | 1866915408462741504 |
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| author | Călugăreanu, Grigore Pop, Horia F. Vasiu, Adrian |
| author_facet | Călugăreanu, Grigore Pop, Horia F. Vasiu, Adrian |
| contents | A unimodular $2\times 2$ matrix with entries in a commutative $R$ is called extendable (resp.\ simply extendable) if it extends to an invertible $3\times 3$ matrix (resp.\ invertible $3\times 3$ matrix whose $(3,3)$ entry is $0$). We obtain necessary and sufficient conditions for a unimodular $2\times 2$ matrix to be extendable (resp.\ simply extendable) and use them to study the class $E_2$ (resp.\ $SE_2$) of rings $R$ with the property that all unimodular $2\times 2$ matrices with entries in $R$ are extendable (resp.\ simply extendable). We also study the larger class $Π_2$ of rings $R$ with the property that all unimodular $2\times 2$ matrices of determinant $0$ and with entries in $R$ are (simply) extendable (e.g., rings with trivial Picard groups or pre-Schreier domains). Among Dedekind domains, polynomial rings over $\mathbb Z$ and Hermite rings, only the EDRs belong to the class $E_2$ or $SE_2$. If $as(R)\le 2$, then $R$ is an $E_2$ ring iff it is an $SE_2$ ring. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_05780 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matrix invertible extensions over commutative rings. Part I: general theory Călugăreanu, Grigore Pop, Horia F. Vasiu, Adrian Commutative Algebra 15A83, 13G05, 19B10, 13A05, 13F05, 15B33 A unimodular $2\times 2$ matrix with entries in a commutative $R$ is called extendable (resp.\ simply extendable) if it extends to an invertible $3\times 3$ matrix (resp.\ invertible $3\times 3$ matrix whose $(3,3)$ entry is $0$). We obtain necessary and sufficient conditions for a unimodular $2\times 2$ matrix to be extendable (resp.\ simply extendable) and use them to study the class $E_2$ (resp.\ $SE_2$) of rings $R$ with the property that all unimodular $2\times 2$ matrices with entries in $R$ are extendable (resp.\ simply extendable). We also study the larger class $Π_2$ of rings $R$ with the property that all unimodular $2\times 2$ matrices of determinant $0$ and with entries in $R$ are (simply) extendable (e.g., rings with trivial Picard groups or pre-Schreier domains). Among Dedekind domains, polynomial rings over $\mathbb Z$ and Hermite rings, only the EDRs belong to the class $E_2$ or $SE_2$. If $as(R)\le 2$, then $R$ is an $E_2$ ring iff it is an $SE_2$ ring. |
| title | Matrix invertible extensions over commutative rings. Part I: general theory |
| topic | Commutative Algebra 15A83, 13G05, 19B10, 13A05, 13F05, 15B33 |
| url | https://arxiv.org/abs/2404.05780 |