SIGMA 4 (2008), 072, 7 pages arXiv:0807.1790
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
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.
non-unimodular geometry over rings; smallest ring of ternions; Fano plane.
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