We introduce the notions of Jacobson radical of a Γ−semiring and semisimple Γ−semiring and characterize them via operator semirings. AMS Mathematics Subject Classification (2000): 16Y60, 16Y99, 20N10

Problem 2. Let A be a central division algebra (of finite dimension) over a field k. Let [A, A] be the k-subspace of A spanned by the elements ab − ba with a, b ∈ A. Show that [A, A] = A. Solution. Let K be the algebraic closure of k, and consider B = A ⊗ k K. Then B ∼ = M n (K) for some n ∈ N, and thus we can understand [B, B] ∼ = [A, A] ⊗ k K. In this case, [B, B] contains only matrices of tr...

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 accor...