Simple artinian rings as sums of nilpotent subrings

Decompositions of simple artinian rings as additive sums of nilpotent subrings are considered. In particular, necessary and sufficent conditions for minimal decompositions are found in terms of the underlying division ring. This is used to prove that any algebra with 1 can be unitarily embedded into a simple algebra with 1 which is a sum of four subalgebras with zero multiplication and also into a simple algebra which is a sum of three nilpotent subalgebras of degree 3. Since our proof is constructive and is based on simple artinian rings, the latter result can be viewed as an extension and a strengthened version of Bokut's theorem, 2].