= Primary submodule
Over a <commutative ring> $R$, a proper <submodule> $N\subsetneq M$ is primary when $rm\in N$ and $m\notin N$ imply $r^kM\subseteq N$ for some positive integer $k$. Equivalently, each scalar that kills a nonzero element of $M/N$ acts nilpotently on the entire quotient. Its radical annihilator is a <prime ideal>.
For example, $p^r\mathbb Z$ is a primary submodule of the integer module $\mathbb Z$ when $p$ is prime and $r\geq1$: if a scalar kills a nonzero residue modulo $p^r$, it is divisible by $p$, and a sufficiently high power kills all residues. By contrast, $6\mathbb Z$ is not primary: $3\cdot2\in6\mathbb Z$, but $2\notin6\mathbb Z$ and no power of three kills $\mathbb Z/6\mathbb Z$.
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