= Associated prime of a module
{title2=$\operatorname{Ass}_R(M)$}
= Associated primes of a module
{synonym}
= Associated prime
{synonym}
An associated prime of an $R$-<module> $M$ is a <prime ideal> equal to the <annihilator> $\operatorname{Ann}_R(m)$ of some nonzero $m\in M$. The set of these primes is denoted $\operatorname{Ass}_R(M)$. Over a <Noetherian ring>, any maximal member among the <annihilators> of nonzero elements is prime: if $abm=0$ and $bm\ne0$, maximality forces $\operatorname{Ann}_R(bm)=\operatorname{Ann}_R(m)$, so $am=0$. Hence every nonzero <module> has an associated prime.
Back to article page
