Primary ideal

Summary

A primary ideal of a commutative ring is a proper ideal $Q$ with this property. Whenever a product $ab$ lies in $Q$ and $a$ does not, some power of $b$ lies in $Q$. Primary ideals generalise powers of prime numbers: in $\Z$ the nonzero primary ideals are the ideals $(p^n)$ for primes $p$. The radical of a primary ideal is a prime ideal. Primary ideals are the building blocks of primary decomposition.