Nilradical of subring lemma

Statement
Suppose $$R$$ is a unital subring of a commutative unital ring $$S$$. Then, the nilradical of $$R$$ equals the intersection of $$R$$ with the nilradical of $$S$$.

Applications

 * When both $$R$$ and $$S$$ are Jacobson rings (for instance, when they are both finitely generated algebras over a field) then for both rings, the Jacobson radical equals the nilradical. Thus, we obtain that the Jacobson radical of $$R$$ equals the intersection of $$R$$ with the Jacobson radical of $$S$$
 * Effect of ideal contraction on Galois correspondent