Answer to Question #24906 in Abstract Algebra for Mohammad

Question

Let Rad R denote one of the two nilradicals, or the Jacobson radical, or the Levitzki radical of R.
For any ideal J ⊆ R such that Rad (R/J) = 0, show that J ⊇ Rad R.

Answer 1

Consider any ideal J ⊆R such that L-rad (R/J) = 0. The image of L-rad R in R/J is still locally
nilpotent, so it must be zero, as
L-rad (R/J) = 0. Thus, we must have L-rad R ⊆ J.