Primary submodules and primary ideals

Let $R$ be a commutative ring, let $M$ be a left $R$-module, and let $Q⊊M$ be a proper submodule. Then $Q$ is primary when every zero divisor on the quotient module $M\mathrm{/}Q$ acts nilpotently on $M\mathrm{/}Q$; equivalently, whenever multiplication by $a\in R$ on $M\mathrm{/}Q$ has nontrivial kernel, there exists $n\ge 1$ with $a^n(M\mathrm{/}Q)=0.$

If $\sqrt{{\mathrm{Ann}}_R(M\mathrm{/}Q)}=\mathfrak{p},$ then $Q$ is called $\mathfrak{p}$-primary.

The radical here is well-defined: ${\mathrm{Ann}}_R(M\mathrm{/}Q)$ is an ideal of $R$. Indeed, it contains $0$, is closed under subtraction, and if $a$ annihilates $M\mathrm{/}Q$ then so does $ra$ for every $r\in R$, by the module axioms.

When $M=R$, viewed as its regular module, a primary submodule is a primary ideal.