Radical ideal

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

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

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)

Relation with other properties

Stronger properties

Incomparable properties

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.

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 $S$ .