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Article Dans Une Revue SIGMA Année : 2008

A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

Résumé

The Fano-Snowflake, a specific $non$-unimodular projective lattice configuration associated with the smallest ring of ternions $R_{\diamondsuit}$ (arXiv:0803.4436 and 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 $R_{\diamondsuit}^{3}$ are 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 (6 -- 7 -- 3) 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 (3 -- 3 -- 1).
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Dates et versions

hal-00295016 , version 1 (11-07-2008)
hal-00295016 , version 2 (27-10-2008)

Identifiants

Citer

Metod Saniga, Petr Pracna. A Jacobson Radical Decomposition of the Fano-Snowflake Configuration. SIGMA, 2008, 4, 072, 7 p. ⟨10.3842/SIGMA.2008.072⟩. ⟨hal-00295016v2⟩

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