How to define a $P$-primary component of $M'\ne \{0\}$ in $M$?

Question

The definition of a $P$-primary component of $0$ in $M$ is given here. But I haven't found a definition of a $P$-primary component of a nonzero submodule $M'$ in $M$. Is there such a notion? If so, what's its definition in terms of the terminology in the question cited above?

Answer 1

From Bourbaki, Commutative Algebra, Ch. IV Associated Prime Ideals and Primary Decomposition, §2:

Let $A$ be a noetherian ring, $M$ an $A$-module, $N$ a submodule of $M$. We call $\,$ primary decomposition of $N$ in $M$ a finite family $(Q_i)_{i\in I}$ of submodules of $M$, primary w.r.t. $M$, and such that $\;N = \bigcap\limits_{i\in I} Q_i$.

$Q$ is said to be primary w.r.t. $M$ if $\:\operatorname{Ass}M/Q$ contains only one element.