Additive decomposition of ideals

Abstract

In this paper, we investigate decomposition of (one-sided) ideals of a unital ring [Formula: see text] as a sum of two (one-sided) ideals, each being idempotent, nil, nilpotent, T-nilpotent, or a direct summand of [Formula: see text]. Among other characterizations, we prove that in a polynomial identity ring every (one-sided) ideal is a sum of a nil (one-sided) ideal and an idempotent (one-sided) ideal if and only if the Jacobson radical [Formula: see text] of [Formula: see text] is nil and [Formula: see text] is von Neumann regular. As a special case, these conditions for a commutative ring [Formula: see text] are equivalent to [Formula: see text] having zero Krull dimension. While assuming Köthe’s conjecture in several occasions to be true, we also raise a question, the affirmative answer to which leads to the truth of the conjecture.