In mathematics, specifically commutative algebra, a proper ideal of a commutative ring is said to be primary. For example, in the ring of integers is a prime number.

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. This result is known as the Lasker–Noether theorem. Consequently, an irreducible ideal of a Noetherian ring is primary.

Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.

Examples and properties

• , the radical

• is primary if and only if every zero divisor is actually zero.)

• Any prime ideal is primary, and moreover an ideal is prime if and only if it is primary and semiprime (also called radical ideal in the commutative case).

• Every primary ideal is primal.

is a primary ideal, then the radical, and this ideal is called the associated prime ideal is said to be -primary.

-primary.

• An ideal whose radical is maximal, however, is primary.

• isth symbolic power.

• .

• is a Noetherian ring to the localization-primary ideals.

• -primary but an infinite product (Krull intersection theorem.

Decomposition of ideals into primary ideals

The primary decomposition of ideals by the Lasker–Noether theorem may be seen as a generalization of the fundamental theorem of arithmetic and other unique factorization domains, to general Noetherian rings. While the unique factorization of elements of a ring into the product of irreducible elements (up to units and reordering) fails in the general case, the Lasker–Noether theorem states that the ideals of a Noetherian ring do still have a type of "unique factorization": any ideal in a Noetherian ring can be written as an intersection of primary ideals of the ring in a primary decomposition, and while these component primary ideals are not necessarily unique, the radicals of these components, the associated primes of the ideal, are unique up to reordering:

Footnotes

References

• On primal ideals, Ladislas Fuchs

External links