Radical of an ideal

$ 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 $.

Comments

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 $.

Radical ideal is the phrase sometimes used to denote an ideal that is equal to its radical.

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.

References

[a1]	P.M. Cohn, "Algebra" , II , Wiley (1977) pp. Sects. 11.2, 11.10
[a2]	A. Grothendieck, "Eléments de géométrie algébrique. I: Le language des schémas" Publ. Math. IHES , 20 (1960) pp. 80