= Solution
In the terminology of a <minimal primary decomposition>, an <associated prime of a primary decomposition> of $I$ is a <prime ideal> occurring as $\sqrt{\mathfrak q_j}$ in such a decomposition. The <first uniqueness theorem for primary decomposition> says that this collection is independent of the chosen <minimal primary decomposition>. Among these <prime ideals>, an <isolated prime of a primary decomposition> is minimal under inclusion, equivalently a <minimal prime over an ideal> $I$; an embedded <prime ideal> strictly contains another member of the collection. Thus
$$
\boxed{\text{isolated}=\text{minimal over }I,\qquad\text{embedded}=\text{not minimal over }I.}
$$
When $R$ is a <Noetherian ring>, these are precisely the <associated primes of a module> $R/I$. In that case the equivalent definition is
$$
\mathfrak p=\operatorname{Ann}_R(x+I)=(I:x)
\quad\text{for some }x\notin I,
$$
where $(I:x)$ is a <colon ideal>. The decomposition terminology applies when a <minimal primary decomposition> exists; a general <ring> need not provide one for every <ideal>.
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