Nilradical of subring lemma

This article is about the statement of a simple but indispensable lemma in commutative algebra
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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