TITLE: Koethe  conjecture
QUESTION [7 upvotes]: First  of  all, has  the  Koethe  conjecture  been  solved ?! Does a proof of  the  following  statement  really  imply  the  proof  of  the  conjecture ?

The sum of two right nil ideals in any ring is nil.

Are there  other  equivalent  statements  that  imply  the  conjecture ?
Thanks, SB

REPLY [3 votes]: Not solved. Many special cases though. The paper cited above looks nice, and another short survey is given in Lam's First Course in Noncommutative rings, around page 164. There are a bunch of equivalent statements there, but maybe the above paper covers them all.