Radically closed subring

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This article defines a property that can be evaluated for a unital subring in a commutative unital ring: given any commutative unital ring and a subring thereof, the property is either true or false for the pair
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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 $n\geq 1$ for which $x^{n}\in S$, we have $x\in S$.

Relation with other properties

Stronger properties

• Integrally closed subring
• 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. For full proof, refer: Equivalence of definitions of radical of an ideal

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