Self-injective von Neumann regular rings and Kothe's conjecture

One of the many equivalent formulation of the Kothe's conjecture is the assertion that there exists no unital ring which contains two nil right ideals whose sum is not nil. We discuss several consequences of an observation that if the Koethe conjecture fails then there exists a counterexample in the form of a countable local subring of a suitable self-injective prime von Neumann regular unital ring. (C) 2020 Elsevier B.V. All rights reserved.