primary ideal

Failing that, an ideal may at least factor into the intersection of primary ideals.

The radical of a primary ideal is prime.

A semiprime primary ideal is prime.

Tertiary ideals generalize primary ideals to the case of noncommutative rings.

Every primary ideal is tertiary.

Every primary ideal of a principal ideal domain is an irreducible ideal.

However, primary ideals which are associated with non-minimal prime ideals are in general not unique.

Tertiary ideals and primary ideals coincide for commutative rings.

Every ideal J (through primary decomposition) is expressible as a finite intersection of primary ideals.

Noether gave an example of a non-commutative Noetherian ring with a right ideal that is not an intersection of primary ideals.

Various methods of generalizing primary ideals to noncommutative rings exist but the topic is most often studied for commutative rings.

Note that in a PID, primary ideals are powers of primes, so the elementary divisors .

For the special case of ideals it states that every ideal of a Noetherian ring is a finite intersection of primary ideals.

Thus, a primary decomposition of (n) corresponds to representing (n) as the intersection of finitely many primary ideals.

In the case of non-commutative rings, the class of tertiary ideals is a useful substitute for the class of primary ideals.

To begin with, it is clear that prime ideals are semiprime, and that for commutative rings, a semiprime primary ideal is prime.

But the impact of the primary group is so great that individuals cling to primary ideals in more complex associations and even create new primary groupings within formal organizations.

Another important milestone was the work of Hilbert's student Emanuel Lasker, who introduced primary ideals and proved the first version of the Lasker-Noether theorem.

Let R be a Noetherian ring, and I an ideal in R. Then I has an irredundant primary decomposition into primary ideals.

More over, this decomposition is unique in the following sense: the set of associated prime ideals is unique, and the primary ideal above every minimal prime in this set is also unique.

The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals.

Specifically, if an ideal J is decomposed as a finite intersection of primary ideals, it is known that the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with .

In mathematics, the Lasker-Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals).