Geodesic
A Jacobson Radical Decomposition of the Fano-Snowflake Configuration
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

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The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions $R_{\diamondsuit}$ (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of $R_{\diamondsuit}$. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left $R_{\diamondsuit}$-module $R_{\diamondsuit}^3$ 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.
Keywords: non-unimodular geometry over rings; smallest ring of ternions; Fano plane. 
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Metod Saniga; Petr Pracna. A Jacobson Radical Decomposition of the Fano-Snowflake Configuration. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a71/

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