One of the many equivalent formulation Köthe's conjecture is assertion that there exists no unital ring which contains two nil right ideals whose sum not nil. We discuss several consequences an observation if Koethe fails then a counterexample in form countable local subring suitable self-injective prime von Neumann regular ring.

Let (R,m, k) be a noetherian local ring. It is well-known that R is regular if and only if the injective dimension of k is finite. In this paper it is shown that R is Gorenstein if and only if the Gorenstein injective dimension of k is finite. On the other hand a generalized version of the so-called Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension. ...