Ideal with prime radical

Definition

An ideal in a commutative unital ring is said to have prime radical if its radical is a prime ideal.

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
prime ideal	product of two elements in the ideal implies at least one of them is in the ideal	follows from the fact that prime implies radical, so a prime ideal equals its own radical	prime radical not implies prime	click here
primary ideal	product of two elements in the ideal implies either the first or a power of the second is in the ideal	primary implies prime radical	prime radical not implies primary	click here
power of a prime ideal	product of a prime ideal with itself finitely many times
ideal with maximal radical	radical is a maximal ideal