Spreadable nilpotent nonassociative rings and \(K\)-algebras (Q1306491)

Let \(R\) be a nilpotent, not necessarily associative ring. Assume \(R\) admits a spread, i.e., there exists a family of subrings \(S\), with \(|S|\geq 3,\) covering \(R,\) such that \(U_1\cap U_2=\{0\}\), \(U_1+U_2=R\) for all \(U_1,U_2\in S.\) The author proves that then the multiplication of \(R\) is trivial, i.e., \(ab=0\) for all \(a,b\in R.\) This naturally applies to \(K\)-algebras where the spread elements form vector subspaces, as well.