I don't understand the definition of minimal (isolated) prime ideals, quoted from Atiyah-Macdonald's "Introduction to Commutative Algebra":
Let $\mathfrak{a}$ be a decomposable ideal with minimal decomposition $$ \mathfrak{a} =\bigcap_{i\leq n} \mathfrak{q}_i $$ Let $ \mathfrak{p}_i = r(\mathfrak{q}_i)$
$\dots$ (Example here)
The minimal elements of the set $\{\mathfrak{p}_i\}$ are called the minimal or isolated prime ideals belonging to $\mathfrak{a}$.
In the example mentioned above, the authors provides us with a case where $$ \mathfrak{p}_1 \subset \mathfrak{p}_2$$ Speaking of minimality in that context is somewhat confusing to me; Do we want the minimal set with respect to inclusion (i.e. take $\mathfrak{p}_1$ above, ditch $\mathfrak{p}_2$)? Or rather take $\mathfrak{p}_2$ since it already "covers" $\mathfrak{p}_1$?
Minimal, or isolated, for the primes $\mathfrak{p}_i$ means minimal with respect to inclusion, so in your example, $\mathfrak{p}_1$ is minimal (isolated), but $\mathfrak{p}_2$ is not.