Primary decompositions, minimality, and isolated components

Let $R$ be a commutative ring, let $M$ be a left $R$-module, and let $N\subseteq M$ be a submodule.

A primary decomposition of $N$ in $M$ is an expression $N=Q_1\cap \cdots \cap Q_r$ with each $Q_i⊊M$ a primary submodule.

Such a decomposition is minimal when:

1. no component is redundant, so omitting any $Q_i$ changes the intersection;
2. the radicals $\sqrt{{\mathrm{Ann}}_R(M\mathrm{/}{Q}_i)}$ are pairwise distinct.

These radicals are well-defined because Primary submodules and primary ideals establishes that each module annihilator ${\mathrm{Ann}}_R(M\mathrm{/}{Q}_i)$ is an ideal of $R$.

In a minimal decomposition, a component $Q_i$ is isolated when its radical is minimal, under inclusion, among the radicals occurring in the decomposition.