A Jacobson Radical Decomposition of the Fano-Snowflake Configuration
The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions Rà (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of Rà. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the fre...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2008 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2008
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149008 |
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| Cite this: | A Jacobson Radical Decomposition of the Fano-Snowflake Configuration / M. Saniga, P. Pracna // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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Saniga, M. Pracna, P. 2019-02-19T12:55:50Z 2019-02-19T12:55:50Z 2008 A Jacobson Radical Decomposition of the Fano-Snowflake Configuration / M. Saniga, P. Pracna // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 8 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 51C05; 51Exx https://nasplib.isofts.kiev.ua/handle/123456789/149008 The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions Rà (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of Rà. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left Rà-module Rà³ is shown to split into three disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson radical entries in the generating vector is 2, 1 or 0, respectively. The corresponding ''ternion-induced'' factorization of the lines of the Fano plane sitting in the middle of the Fano-Snowflake is found to differ fundamentally from the natural one, i.e., from that with respect to the Jacobson radical of the Galois field of two elements. The work was partially supported by the VEGA grant agency projects Nos. 6070 and 7012, the CNRS-SAV Project No. 20246 and by the Action Austria–Slovakia project No. 58s2. We thank Hans Havlicek (Vienna University of Technology) for valuable comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Jacobson Radical Decomposition of the Fano-Snowflake Configuration Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
A Jacobson Radical Decomposition of the Fano-Snowflake Configuration |
| spellingShingle |
A Jacobson Radical Decomposition of the Fano-Snowflake Configuration Saniga, M. Pracna, P. |
| title_short |
A Jacobson Radical Decomposition of the Fano-Snowflake Configuration |
| title_full |
A Jacobson Radical Decomposition of the Fano-Snowflake Configuration |
| title_fullStr |
A Jacobson Radical Decomposition of the Fano-Snowflake Configuration |
| title_full_unstemmed |
A Jacobson Radical Decomposition of the Fano-Snowflake Configuration |
| title_sort |
jacobson radical decomposition of the fano-snowflake configuration |
| author |
Saniga, M. Pracna, P. |
| author_facet |
Saniga, M. Pracna, P. |
| publishDate |
2008 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions Rà (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of Rà. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left Rà-module Rà³ is shown to split into three disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson radical entries in the generating vector is 2, 1 or 0, respectively. The corresponding ''ternion-induced'' factorization of the lines of the Fano plane sitting in the middle of the Fano-Snowflake is found to differ fundamentally from the natural one, i.e., from that with respect to the Jacobson radical of the Galois field of two elements.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149008 |
| citation_txt |
A Jacobson Radical Decomposition of the Fano-Snowflake Configuration / M. Saniga, P. Pracna // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 8 назв. — англ. |
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| first_indexed |
2025-12-07T16:24:36Z |
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2025-12-07T16:24:36Z |
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