Answer to Question #25247 in Abstract Algebra for Mohammad
Show that from Kothe’s Conjecture (“The sum of two nil left ideals in any ring is nil”.) followsthe statement: if I is a nil ideal in any ring R, then Mn(I) is nil for any n.
Note that Mn(I) equals sum of all Jkwhere Jk is the left ideal in Mn(R) consisting of matrices with k-thcolumn entries from I and all other entries zero. Then, each Jk isnil, so by assumption (and induction), Mn(I) is nil.