Definition 1

*Let $A$ be a Noetherian ring, $N$ an $A$-module and $Q$ a submodule of $N$. If the module $M = N/Q$ satisfies the conditions of Proposition 1, $Q$ is called p-primary with respect to $N$ (or in $N$).

When there is no ambiguity, we shall simply say that $Q$ is "p-primary" or "primary"; clearly for every submodule $N' \neq Q$ of $N$ containing $Q$, $Q$ is p-primary in $N'$.

Definition 1 applies in particular to the case $N = A$; the submodules of $N$ are then the ideals of $A$ and hence an ideal $q$ of $A$ is called primary if $\operatorname{Ass}(A/q)$ has a single element or, what amounts to the same, if $A \neq q$ and every divisor of zero in the ring $A/q$ is nilpotent. If $q$ is p-primary, it follows from Definition 1 that $p$ is the radical (Chapter II, $§ 2$, no. 6) of the ideal $q$.