Is there an overview of rings for which the Köthe conjecture is known to hold? In particular, I am interested in endomorphism rings of graded modules over multivariate polynomial rings. This survey states that the conjecture holds for left artinian rings (alas, without a reference). However, in the notion of that survey, to hold means that "the ideal generated by every nil left ideal of R is nil."
So my explicit question is: In any left artinian ring, or in a particular special case thereof, is it true that the sum of two left nil ideals is nil again?
Yes, because in a left Artinian ring nil left ideals are nilpotent, and the sum of two nilpotent left ideals is always nilpotent.