A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions $R_{\diamondsuit}$ (arXiv:0803.4436 and arXiv: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 $R_{\diamondsuit}^3$ 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.