The Geometric Intuition behind Minimal Primary Decompositions versus an arbitrary Primary Decomposition

Given an ideal in a Noetherian domain, it has a primary decomposition. This decompositions may not be minimal. I am interested in the relationship of primary ideals that are not in a minimal primary decomposition in relation to those that are.

Even more specifically, I have the following problem. I have a radical ideal $Q$ and a set of minimal primes $P_1,\dots,P_n$ containing $Q$. Of course $Q=\bigcap{P_i}$, but this decompositions is not necessarily minimal. Let $I\subset[n]$ be such that $Q=\bigcap_{k\in I}{P_{i_k}}$ is a minimal decomposition

Each minimal prime is a maximal irreducible component of $V(Q)$, however, it seems that if $I\subsetneq[n]$, then for some $j\notin I$, $P_j$ is an maximal irreducible component contained in the union $\bigcup_{k\in I}{V(P_{i_k})}$, but I don't see how such a thing can be possible.

Any clarification of the geometric situation would be greatly appreciated.

• I am confused. For radical ideal $Q$, the decomposition $Q=\bigcap P_i$ is always minimal, isn't it? – t3suji Sep 21 '15 at 20:15
• Oh my goodness, I believe you're correct. I must have lost my mind. This clears up that problem then. – batconjurer Sep 22 '15 at 9:18