**Primary Ideal**

In mathematics, specifically commutative algebra, a proper ideal *Q* of a commutative ring *A* is said to be **primary** if whenever *xy* is an element of *Q* then *x* or *y**n* is also an element of *Q*, for some *n>0*. For example, in the ring of integers **Z**, (*p*n) is a primary ideal if *p* 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 Laskerâ€“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.

Read more about Primary Ideal: Examples and Properties

### Other articles related to "primary ideal, ideal, primary":

**Primary Ideal**- Examples and Properties

... Any prime

**ideal**is

**primary**, and moreover an

**ideal**is prime if and only if it is

**primary**and semiprime ... Every

**primary ideal**is primal ... If Q is a

**primary ideal**, then the radical of Q is necessarily a prime

**ideal**P, and this

**ideal**is called the associated prime

**ideal**of Q ...

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