Answer to Question #24905 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 I ⊆ Rad R, show that Rad (R/I) = (Rad R)/I.

Answer 1

Finally, we deal with the Levitzkiradical L-rad R. Again, consider an ideal N ⊇ L-rad R with N2 ⊆ L-rad R. We claim that N islocally nilpotent. In fact, if a1, . . . , an ∈ N, form the products aij : = aiaj ∈ L-rad R.There exists an integer m suchthat any product of m elements from the set {aij} is zero.Therefore, any product of 2m elements from the set {ai} is zero.This shows that N ⊆ L-rad R,so L-rad R is semiprime. If I is an ideal ⊆ L-rad R, a similar argument shows that L-rad(R/I) = (L-rad R)/I.