Intersection of prime equals radical

This article gives the statement and possibly, proof, of an implication relation between two commutative unital ring properties. That is, it states that every commutative unital ring satisfying the first commutative unital ring property must also satisfy the second commutative unital ring property
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Statement

An ideal in a commutative unital ring is radical if and only if it is an intersection of prime ideals.

Proof

Because of the way the properties are quotient-determined, we can reduce this to the statement that:

nilradical equals intersection of all prime ideals