Conjugate (nil) clean rings and Köthe’s problem

Question 3 of [3] asks whether the matrix ring Mn(R) is nil clean, for any nil clean ring R. It is shown that, positive answer to this question is equivalent to positive solution for Köthe’s problem in the class of algebras over the field F2. Other equivalent problems are also discussed. The classes of conjugate clean and conjugate nil clean rings, which lie strictly between uniquely (nil) clean and (nil) clean rings are introduced and investigated.