Recall that given an ideal $\mathfrak{a}\subseteq A$ and $x\in A$, we define $(\mathfrak{a}:x)=\{y\in A|xy\in \mathfrak{a}\}$.

Let $\mathfrak{q}\subseteq A$ be $\mathfrak{p}$-primary, and $x\in A$. Then the following hold:

1. If $x\in \mathfrak{q}$, then $(\mathfrak{q}:x)=$1;;$A$
2. If $x\notin \mathfrak{p}$, then $(\mathfrak{q}:x)=$1;;$\mathfrak{q}$
3. If $x\in \mathfrak{p}-\mathfrak{q}$, then $(\mathfrak{q}:x)$ 1;; is $\mathfrak{p}$-primary