Associated prime

Remarks

For an ideal $I$ of $R$, the associated primes of $I$ usually means $\operatorname{Ass}_R(R/I)$. Example: for $M = \Z/12\Z$ over $\Z$, $\operatorname{Ann}(6) = (2)$ and $\operatorname{Ann}(4) = (3)$, so $(2)$ and $(3)$ are associated primes; in fact $\operatorname{Ass}(M) = \set{(2), (3)}$. Non-example: $(0)$ is not an associated prime of $\Z/12\Z$, because $12$ kills every element, so no annihilator is $(0)$. For $M = \Z$, the annihilator of every nonzero element is $(0)$, so $\operatorname{Ass}(\Z) = \set{(0)}$. Equivalently, $P \in \operatorname{Ass}_R(M)$ when $M$ has a submodule isomorphic to $R/P$. For rings that are not Noetherian, some authors use other variants, such as weakly associated primes.