Köthe’s Conjecture: If $Nil^*R =0$, then R has no nonzero nil one-sided ideals. ($Nil^*R$ is the sum of all nil ideals in R)
I am studying T.Y. Lam book and there is 2 equivalent formulations:
(A) Every nil left or right ideal of a ring R is contained in $Nil^*R$.
(B) The sum of two nil left (resp., right) ideals of R is also nil.
I was able to show that (A) $\Rightarrow$ Köthe’s Conjecture and (A) $\Rightarrow $(B). I don’t know how to show Köthe’s Conjecture $\Rightarrow$ (A) and (B) $\Rightarrow$ (A).
Any help would be great.