# Answer to Question #24905 in Abstract Algebra for Mohammad

Question #24905

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.

For any ideal I ⊆ Rad R, show that Rad (R/I) = (Rad R)/I.

Expert's answer

Finally, we deal with the Levitzkiradical

*L*-rad*R*. Again, consider an ideal*N**⊇**L*-rad*R*with*N*2*⊆**L*-rad*R*. We claim that*N*islocally nilpotent. In fact, if*a*1*, . . . , 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 2*m*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.*
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