## Ideal in $k[X,Y]$. Tomorrow exam!

Let $A$ be a commutative ring with 1 and $a$ an ideal of $A$. A prime ideal $p$ is said to be associated to $a$ if there exists an $x in A$ such that $p=r(a:x)$, where $r(a:x)$ is the radical of the quotient of $a$ and $x$.

Show that if $a$ is prymary then there exists precisely one prime ideal of $A$ that is associated to this ideal $a$.

And if $a$ is decomposable and there exists precisely one prime ideal of $A$ that is associated to this ideal $a$, then $a$ is primary.