Self-injective von Neumann regular rings and K\"othe's Conjecture

TL;DR

This paper explores the implications of K"othe's Conjecture in ring theory, suggesting that a counterexample would be a countable local subring within a self-injective von Neumann regular ring, thus connecting conjecture failure to specific ring structures.

Contribution

It introduces a new perspective linking potential counterexamples of K"othe's Conjecture to subrings of self-injective von Neumann regular rings, providing a novel approach to the problem.

Findings

If K"othe's Conjecture fails, a counterexample exists as a countable local subring.

Such a subring would be contained in a self-injective prime von Neumann regular ring.

The paper discusses consequences of this observation for ring theory.

Abstract

One of the many equivalent formulation of the K\"othe's conjecture is the assertion that there exists no 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 ring.