Let $\mathfrak{a}\subseteq A$ be an ideal. Then the following hold:

1. If $\mathfrak{a}\subseteq \mathfrak{p}$ for some $\mathfrak{p}\in Spec(A)$, then there exists a ${\mathfrak{p}}_i\in Ass(A/\mathfrak{a})$ such that $\mathfrak{a}\subseteq {\mathfrak{p}}_i\subseteq \mathfrak{p}$.
2. The isolated primes of $Ass(A/\mathfrak{a})$ are exactly the 1;;minimal primes containing $\mathfrak{a}$.

PROOF:

• Prime avoidance