
SIGMA 4 (2008), 072, 7 pages arXiv:0807.1790
https://doi.org/10.3842/SIGMA.2008.072
A Jacobson Radical Decomposition of the FanoSnowflake Configuration
Metod Saniga ^{a} and Petr Pracna ^{b}
^{a)} Astronomical Institute, Slovak Academy of
Sciences, SK05960 Tatranská Lomnica, Slovak Republic
^{b)} J. Heyrovský Institute of Physical Chemistry,
v.v.i., Academy of Sciences of the Czech Republic, Dolejskova 3, CZ18223 Prague 8, Czech Republic
Received July 14, 2008, in final form October 17, 2008; Published online October 24, 2008
Abstract
The FanoSnowflake, a specific configuration associated with the smallest ring of ternions R_{à} (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of R_{à}. The totality of 21 free cyclic submodules generated by nonunimodular
vectors of the free left R_{à}module R_{à}^{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 ''ternioninduced'' factorization of the lines of the Fano plane sitting in the middle of the FanoSnowflake
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:
nonunimodular geometry over rings; smallest ring of ternions; Fano plane.
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