Definition. Coprimary [local-0]

Let $A$ be a Noetherian ring. A nonzero finitely generated $A$ -module $M$ is coprimary if for all $a \in A$ , the multiplication map (reuse element $a$ to denote it)

a : M \to M \\ x \mapsto ax

is injective or nilpotent.

If $M$ is coprimary, then the set

P := \{ a \in A \mid a \text{ is nilpotent} \}

forms a prime ideal in $A$ .

Proof. [local-1]

To check $P$ is a prime ideal, we want to check that if $a \not\in P$ and $b \not\in P$ then $ab \not\in P$ .

Because $M$ is coprimary, so such $a, b$ are injective, and

(ab)(x) = a(b(x))

the composition of injective maps is injective, hence $ab \not\in P$ . □