I've started to study about minimal primary decompositions and have this problem:
Let $I$ be a proper ideal of a commutative Noetherian ring $R$, and let $I=Q_1\cap\cdots\cap Q_n$ with $\sqrt{Q_i}=P_i$ $(i=1,...,n$) be a minimal primary decomposition of $I$. Suppose that $P_i$ is an embedded prime ideal of $I$.
Prove that there are infinitely many different choices of an alternative $P_i$-primary ideal of $R$ which can be substituted for $Q_i$ in the above decomposition so that the result is still a minimal primary decomposition of $I$.
(This is Ex. 8.36 in Steps in Commutative Algebra by R.Y.Sharp.)
I have no idea to showing "infinitely many", thank for your help.