I have the following problem
Let $R$ be a ring, $A$ be a nil left ideal, $B$ be a nil ideal. Prove that $A+B$ is a left nil ideal. (Nil ideal is an ideal where every element is nilpotent.)
Generally, the sum of two nilpotent elements may not be nilpotent (in the case of non-commutative ring), so a set of nilpotent elements could not be an ideal, so I have the question
What condition could make a set of nilpotent elements to be an ideal?
I see Köthe conjecture at https://en.wikipedia.org/wiki/K%C3%B6the_conjecture is an open problem but my problem gives $A$ is left nil ideal and $B$ is a two-sided nil ideal, so then could my problem be easier?