A little Wedderburn principal theorem (Q1801925)

One of the consequences of the Wedderburn principal theorem is that, if \(R\) is a suitable finite-dimensional algebra, then \(R\) is the direct sum of two rings \(S\) and \(T\) such that \(S\) is semi-simple and the radical of \(T\) is essential as a two-sided ideal of \(T\). It is shown that many rings \(R\) have such a decomposition even if the Wedderburn principal theorem does not apply to \(R\), for instance \(R\) can be any left Artinian ring. This follows from a very general result about the decomposition of a ring with a suitable nil ideal.