Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 072, 7 pages      arXiv:0807.1790      http://dx.doi.org/10.3842/SIGMA.2008.072

A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

Metod Saniga a and Petr Pracna b
a) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
b) J. Heyrovský Institute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejskova 3, CZ-18223 Prague 8, Czech Republic

Received July 14, 2008, in final form October 17, 2008; Published online October 24, 2008

Abstract
The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions R (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of R. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left R-module R3 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.

Key words: non-unimodular geometry over rings; smallest ring of ternions; Fano plane.

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