Automorphisms of finitary incidence rings
Let \(P\) be a quasiordered set, \(R\) an associative unital ring, \(\mathcal{C}(P,R)\) a partially ordered category associated with the pair \((P,R)\)[6], \(FI(P,R)\) a finitary incidence ring of \(\mathcal{C}(P,R)\)[6]. We prove that the group \({\rm Out}FI\) of outer automorphisms of \(FI(P,R)\)...
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| Datum: | 2018 |
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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/632 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543206977568768 |
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| author | Khripchenko, Nikolay |
| author_facet | Khripchenko, Nikolay |
| author_sort | Khripchenko, Nikolay |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-04T09:11:25Z |
| description | Let \(P\) be a quasiordered set, \(R\) an associative unital ring, \(\mathcal{C}(P,R)\) a partially ordered category associated with the pair \((P,R)\)[6], \(FI(P,R)\) a finitary incidence ring of \(\mathcal{C}(P,R)\)[6]. We prove that the group \({\rm Out}FI\) of outer automorphisms of \(FI(P,R)\) is isomorphic to the group \({\rm Out}\mathcal{C}\) of outer automorphisms of \(\mathcal{C}(P,R)\) under the assumption that \(R\) is indecomposable. In particular, if \(R\) is local, the equivalence classes of \(P\) are finite and \(P=\bigcup\limits_{i\in I}P_i\) is the decomposition of \(P\) into the disjoint union of the connected components, then \({\rm Out}FI\cong (H^1(\overline P,C(R)^*)\rtimes\prod\limits_{i\in I}{\rm Out}R)\rtimes{\rm Out}P\). Here \(H^1(\overline P,C(R)^*)\) is the first cohomology group of the order complex of the induced poset \(\overline P\) with the values in the multiplicative group of central invertible elements of \(R\). As a consequences, Theorem 2 [9], Theorem 5 [2 ] and Theorem 1.2 [8] are obtained. |
| first_indexed | 2025-12-02T15:40:30Z |
| format | Article |
| id | admjournalluguniveduua-article-632 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:40:30Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-6322018-04-04T09:11:25Z Automorphisms of finitary incidence rings Khripchenko, Nikolay finitary incidence algebra, partially ordered category,quasiordered set, automorphism 18E05, 18B35, 16S50, 16S60,16G20, 08A35 Let \(P\) be a quasiordered set, \(R\) an associative unital ring, \(\mathcal{C}(P,R)\) a partially ordered category associated with the pair \((P,R)\)[6], \(FI(P,R)\) a finitary incidence ring of \(\mathcal{C}(P,R)\)[6]. We prove that the group \({\rm Out}FI\) of outer automorphisms of \(FI(P,R)\) is isomorphic to the group \({\rm Out}\mathcal{C}\) of outer automorphisms of \(\mathcal{C}(P,R)\) under the assumption that \(R\) is indecomposable. In particular, if \(R\) is local, the equivalence classes of \(P\) are finite and \(P=\bigcup\limits_{i\in I}P_i\) is the decomposition of \(P\) into the disjoint union of the connected components, then \({\rm Out}FI\cong (H^1(\overline P,C(R)^*)\rtimes\prod\limits_{i\in I}{\rm Out}R)\rtimes{\rm Out}P\). Here \(H^1(\overline P,C(R)^*)\) is the first cohomology group of the order complex of the induced poset \(\overline P\) with the values in the multiplicative group of central invertible elements of \(R\). As a consequences, Theorem 2 [9], Theorem 5 [2 ] and Theorem 1.2 [8] are obtained. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/632 Algebra and Discrete Mathematics; Vol 9, No 2 (2010) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/632/166 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | finitary incidence algebra partially ordered category,quasiordered set automorphism 18E05 18B35 16S50 16S60,16G20 08A35 Khripchenko, Nikolay Automorphisms of finitary incidence rings |
| title | Automorphisms of finitary incidence rings |
| title_full | Automorphisms of finitary incidence rings |
| title_fullStr | Automorphisms of finitary incidence rings |
| title_full_unstemmed | Automorphisms of finitary incidence rings |
| title_short | Automorphisms of finitary incidence rings |
| title_sort | automorphisms of finitary incidence rings |
| topic | finitary incidence algebra partially ordered category,quasiordered set automorphism 18E05 18B35 16S50 16S60,16G20 08A35 |
| topic_facet | finitary incidence algebra partially ordered category,quasiordered set automorphism 18E05 18B35 16S50 16S60,16G20 08A35 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/632 |
| work_keys_str_mv | AT khripchenkonikolay automorphismsoffinitaryincidencerings |