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

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)

For non-commutative rings

There are the following definitions:

• Radical ideal (noncommutative rings)
• Semiprime ideal