A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

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 R ♦ 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.