Answer to Question #25247 in Abstract Algebra for Mohammad

Question

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.

Answer 1

Note that Mn(I) equals sum of all J_kwhere J_k is the left ideal in Mn(R) consisting of matrices with k-thcolumn entries from I and all other entries zero. Then, each J_k isnil, so by assumption (and induction), M_n(I) is nil.