Associated prime of a module (${\mathrm{Ass}}_R(M)$)

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

An associated prime of a module $R/I$ is embedded when it is not minimal over $I$. For example, in $k[x,y]/(x^2,xy)$ the nonzero class of $x$ has annihilator $(x,y)$, while the unique minimal prime is $(x)$. Embedded primes record annihilators which are invisible if one retains only the reduced irreducible components.

For a Noetherian ring and a proper ideal $I$, every minimal prime $P$ over $I$ is an associated prime of a module $R/I$. The localized quotient has one prime, so its finitely generated maximal ideal is nilpotent and its socle is nonzero. An element with annihilator $P{R}_P$ can be lifted to the quotient. Clearing denominators for a finite generating set of $P$ gives a nonzero element with annihilator exactly $P$.