Radical ideal

This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra

This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings

This property of an ideal in a ring is equivalent to the property of the quotient ring being a/an: reduced ring | View other quotient-determined properties of ideals in commutative unital rings

Definition

An ideal in a commutative unital ring is termed a radical ideal if it satisfies the following equivalent conditions:

• Whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal
• The quotient ring by the ideal has trivial nilradical (that is, it is a reduced ring)
• It is an intersection of prime ideals

Note that unlike for the definition of prime ideals, we do not require a radical ideal to be proper, so the whole ring is always a radical ideal in itself.

Relation with other properties

Stronger properties

• Maximal ideal
• Prime ideal

Incomparable properties

• Irreducible ideal
• Primary ideal

Metaproperties

Intersection-closedness

This property of ideals in commutative unital rings is intersection-closed: an arbitrary intersection of ideals with this property, also has this property

An arbitrary intersection of radical ideals is again a radical ideal.

Contraction-closedness

This property of ideals in commutative unital rings is contraction-closed: a contraction of an ideal with this property, also has this property
View other contraction-closed properties of ideals in commutative unital rings

If $f:R\to S$ is a homomorphism of commutative unital rings, and $I$ is a radical ideal in $S$, then $f^{-1}(I)$ is a radical ideal in $R$.

Intermediate subring condition

This property of ideals satisfies the intermediate subring condition for ideals: if an ideal has this property in the whole ring, it also has this property in any intermediate subring
View other properties of ideals satisfying the intermediate subring condition

A radical ideal of a ring is also a radical ideal in any intermediate subring. This corresponds to the fact that any subring of a reduced ring is again reduced.

Transfer condition

This property of ideals satisfies the transfer condition for ideals: if an ideal satisfies the property in the ring, its intersection with any subring satisfies the property inside that subring

If $I$ is a radical ideal in $R$, and $S$ is a subring of $R$, then $I\cap S$ is a radical ideal in $S$.