Simple Artinian rings as sums of nilpotent subrings (Q1614987)

Let \(D\) be a division ring. It is proved that the matrix algebra \(M_3(D)\) (respectively \(M_2(D)\)) is a sum of three (respectively four) nilpotent subrings of degree 3 (respectively 2) if and only if there exists \(t\in D\) such that the inner derivation \(\text{ad}_t\colon D\to D\) is ``onto''. It is given an example of a field \(L\) over any field \(k\) with a \(k\)-derivation which is ``onto''. Then the division algebra of skew Laurent formal power series over \(L\) is an example of a field \(D\) mentioned above. As a result it is proved that any algebra \(R\) with 1 over a field \(k\) can be unitarily embedded into a simple algebra with 1 which is additively a sum of three (respectively four) subalgebras which are nilpotent of degree 3 (respectively 2). This is an extension of the reviewer's results [see \textit{L. A. Bokut'}, Algebra Logika 15, No. 2, 117-142 (1976; Zbl 0349.16007)].