Answer to Question #17353 in Abstract Algebra for sanches

Question

If an ideal I ⊆ R is such that R/I is J-semisimple, show that I ⊇ rad R.

Answer 1

The J-semisimplicity of R/I means that the intersection of the maximal left ideals of R containing I is exactly I. It follows that rad R, the intersection of all the maximal left ideals of R, is contained in I.