Radical ideal

Definition for commutative rings

An ideal in a commutative unital ring (or in any commutative 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)

Definition for non-commutative rings

There are the following definitions:

• Radical ideal (noncommutative rings)
• Semiprime ideal