Rings for which the Köthe conjecture holds 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?

Hrefnui9

Hrefnui9

Answered question

2022-09-07

Rings for which the Köthe conjecture holds
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. I read one 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?

Answer & Explanation

Mateo Tate

Mateo Tate

Beginner2022-09-08Added 18 answers

Yes, because in a left Artinian ring nil left ideals are nilpotent, and the sum of two nilpotent left ideals is always nilpotent.

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