# Nilradical of subring lemma

From Commalg

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

View other indispensable lemmata

## Statement

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

## Applications

- When both and 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 equals the intersection of with the Jacobson radical of
- Effect of ideal contraction on Galois correspondent