# Answer to Question #24906 in Abstract Algebra for Mohammad

Question #24906

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.

For any ideal J ⊆ R such that Rad (R/J) = 0, show that J ⊇ Rad R.

Expert's answer

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.

nilpotent, so it must be zero, as

L-rad (R/J) = 0. Thus, we must have L-rad R ⊆ J.

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