Trying to understand a remark from Atiyah and Macdonald's *Introduction to Commutative Algebra* Can someone explain the following remark from page 52 of Atiyah and Macdonald's Introduction to Commutative Algebra book.

The names isolated and embedded come from geometry. Thus if $A=k[x_1,...,x_n]$, where $k$ is a field, the ideal $\mathfrak a$ gives rise to a variety $X \subset k^n$. The minimal primes $p_i$ corresponds to the irreducible components of $X$ and the embedded primes corresponds to subvarieties of these, i.e., varieties embedded in the irreducible components.

Moreover, is the above remark useful in finding primary decomposition of an ideal? How do we use this in finding a decomposition?

Let $\mathfrak a=\mathfrak q_1\cap\cdots\cap\mathfrak q_n$ be a reduced primary decomposition. Then $V(\mathfrak a)=V(\mathfrak q_1)\cup\cdots\cup V(\mathfrak q_n)$. Let $\mathfrak p_i=\sqrt{\mathfrak q_i}$. Now we have $X=V(\mathfrak p_1)\cup\cdots\cup V(\mathfrak p_n)$. If $\mathfrak p_i\subset\mathfrak p_j$, then $V(\mathfrak p_j)\subset V(\mathfrak p_i)$, so an embedded prime gives rise to a subvariety of a minimal (isolated) prime. In fact, $X=\bigcup V(\mathfrak p_i)$ where $\mathfrak p_i$ is running through the minimal primes over $\mathfrak a$.
I doubt that in general this can be useful to find out a primary decomposition since you loose a lot of information when passing from $\mathfrak a$ to $V(\mathfrak a)$.
