Let be a ring. An ideal of is a semiprime ideal if it satisfies the following equivalent conditions:

(a) can be expressed as an intersection of prime ideals of ;

(b) if , and , then ;

(c) if is a two-sided ideal of and , then as well;

(d) if is a left ideal of and , then as well;

(e) if is a right ideal of and , then as well.

Here is the product of ideals .

The ring itself satisfies all of these conditions (including being expressed as an intersection of an empty family of prime ideals) and is thus semiprime.

A ring is said to be a semiprime ring if its zero ideal is a semiprime ideal.

Note that an ideal of is semiprime if and only if the quotient ring is a semiprime ring.