**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*. In this situation,*Q*is said to be.*P*-primary

- If
*P*is a maximal prime ideal, then any ideal containing a power of*P*is*P*-primary. Not all*P*-primary ideals need be powers of*P*; for example the ideal (*x*,*y*2) is*P*-primary for the ideal*P*= (*x*,*y*) in the ring*k*, but is not a power of*P*.

- In general powers of a prime ideal
*P*need not be*P*-primary. (An example is given by taking*R*to be the ring*k*/(*xy*−*z*2), with*P*the prime ideal (*x*,*z*). If*Q*=*P*2, then*xy*∈*Q*, but*x*is not in*Q*and*y*is not in the radical*P*of*Q*, so*Q*is not*P*-primary.) However every ideal*Q*with radical*P*is contained in a smallest*P*-primary ideal, consisting of all elements*a*such that*ax*is in*Q*for some*x*not in*P*. In particular there is a smallest*P*-primary ideal containing*P**n*, called the*n*th**symbolic power**of*P*.

- If
*A*is a Noetherian ring and*P*a prime ideal, then the kernel of, the map from*A*to the localization of*A*at*P*, is the intersection of all*P*-primary ideals.

Read more about this topic: Primary Ideal

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