Radically closed subring

Definition
A unital subring $$S$$ of a commutative unital ring $$R$$ is termed radically closed if for every $$x \in R$$ such that there exists $$n \ge 1$$ for which $$x^n \in S$$, we have $$x \in S$$.

Stronger properties

 * Weaker than::Integrally closed subring
 * Weaker than::Algebraically closed subring

Related properties
A radical ideal is an ideal with the analogous property: if $$x^n$$ is in the ideal, so is $$x$$. The radical of an ideal is the smallest ideal containing it that is a radical ideal. It turns out that for any ideal, every element in its radical has the property that some power of it is in the ring.

Thus, if an ideal is contained in a radically closed subring, the radical of that ideal is also contained in that subring.