Koethe conjecture

The Koethe Conjecture is the statement that for any pair of nil right ideals $A$ and $B$ in any ring $R$, the sum $A+B$ is also nil.

If either of $A$ or $B$ is a two-sided ideal, it is easy to see that $A+B$ is nil. (See properties of nil and nilpotent ideals.)

In particular, this means that the Koethe conjecture is true for commutative rings.

It has been shown to be true for many classes of rings, but the general statement is still unproven, and no counter example has been found.