Semiprime ring

is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but

For a commutative ring R, a proper ideal A is a semiprime ideal if A satisfies either of the following equivalent conditions: The latter condition that the complement is "closed under powers" is analogous to the fact that complements of prime ideals are closed under multiplication.

As with prime ideals, this is extended to noncommutative rings "ideal-wise".

A nonempty subset S of a ring R is called an n-system if for any s in S, there exists an r in R such that srs is in S. With this notion, an additional equivalent point may be added to the above list: The ring R is called a semiprime ring if the zero ideal is a semiprime ideal.

In the commutative case, this is equivalent to R being a reduced ring, since R has no nonzero nilpotent elements.

In the noncommutative case, the ring merely has no nonzero nilpotent right ideals.

The inclusion above is sometimes proper in the general case, but for commutative rings it becomes an equality.

and also called Baer's lower nilradical or the Baer-Mccoy radical or the prime radical of R. A right Goldie ring is a ring that has finite uniform dimension (also called finite rank) as a right module over itself, and satisfies the ascending chain condition on right annihilators of its subsets.

Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients.

The Artin–Wedderburn theorem then completely determines the structure of this ring of quotients.