##### Actions

$A$ in a commutative associative ring $R$

The set of all elements $b \in R$ some power of which is contained in $A$. This set is denoted by $\sqrt A$. It is an ideal in $R$; moreover, $\sqrt A \supset A$ and $\sqrt {\sqrt A } = \sqrt A$.

A generalization of this idea is that of the radical of a submodule. Let $M$ be a module over $R$ and let $N$ be a submodule of $M$. The radical of the submodule $N$ is the set of all elements $a \in R$ such that $a ^ {n} M \subset N$ for some integer $n$( in general, depending on $a$). The radical of a submodule is an ideal in $R$.

Consider the quotient ring $R/A$ and the natural quotient homomorphism $\pi : R \rightarrow R/A$. The radical of $A$ is the inverse image of the nil radical (cf. Nil ideal) of $R/A$.
Let $k$ be an algebraically closed field. To each ideal $A \subset R [ X _ {1} \dots X _ {n} ]$ one associates the algebraic set $V( A) \subset k ^ {n}$, $V( A) = \{ {a = ( a _ {1} \dots a _ {n} ) \in k ^ {n} } : {f( a) = 0 \textrm{ for all } f \in A } \}$. The Hilbert Nullstellensatz says that $\{ {g \in R } : {g( x) = 0 \textrm{ for all } x \in V( A) } \} = \sqrt A$. Thus there is in this setting a bijective correspondence between radical ideals and algebraic sets.
In the setting of affine schemes (cf. Affine scheme) $\mathop{\rm Spec} ( R)$ this takes the following form. To each ideal $A \subset R$ one associates the closed subspace $\mathop{\rm Spec} ( R/A) = V( A) = \{ {\mathfrak p } : {\mathfrak p \textrm{ is a "prime" ideal such that } \mathfrak p \supset A } \}$. Conversely, to each closed subspace $V \subset \mathop{\rm Spec} ( R)$ one associates the ideal $I( V)= \{ {f \in R } : {f \in \mathfrak p \textrm{ for all } \mathfrak p \in V } \}$. Then again $IV( A) = \sqrt A$ because $\sqrt A$ is the intersection of all prime ideals containing $A$, and, again, $I$ and $V$ set up a bijective correspondence between radical ideals and closed subsets of $\mathop{\rm Spec} ( R)$. The difference with the setting of the Nullstellensatz is that in that case only prime ideals of the form $( X _ {1} - a _ {1} \dots X _ {n} - a _ {n} )$ are considered.