Biserial minor degenerations of matrix algebras over a field
Let \(n\geq 2\) be a positive integer, \(K\) an arbitrary field, and \( q = [ q ^{(1)}| \ldots | q ^{(n)}]\) an \(n\)-block matrix of \(n\times n\) square matrices \( q ^{(1)}, \ldots, q ^{(n)}\) with coefficients in \(K\) satisfying the conditions (C1) and (C2) listed in the introduction. We study...
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| Datum: | 2018 |
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| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
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| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/636 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Zusammenfassung: | Let \(n\geq 2\) be a positive integer, \(K\) an arbitrary field, and \( q = [ q ^{(1)}| \ldots | q ^{(n)}]\) an \(n\)-block matrix of \(n\times n\) square matrices \( q ^{(1)}, \ldots, q ^{(n)}\) with coefficients in \(K\) satisfying the conditions (C1) and (C2) listed in the introduction. We study minor degenerations \(\mathbb{M}^q_n(K)\) of the full matrix algebra \(\mathbb{M}_n(K)\) in the sense of Fujita-Sakai-Simson [7]. A characterisation of all block matrices \( q = [ q ^{(1)}| \ldots | q ^{(n)}]\) such that the algebra \(\mathbb{M}^q_n(K)\) is basic and right biserial is given in the paper. We also prove that a basic algebra \(\mathbb{M}^q_n(K)\) is right biserial if and only if \(\mathbb{M}^q_n(K)\) is right special biserial. It is also shown that the \(K\)-dimensions of the left socle of \(\mathbb{M}^q_n(K)\) and of the right socle of \(\mathbb{M}^q_n(K)\) coincide, in case \(\mathbb{M}^q_n(K)\) is basic and biserial. |
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