Can intersection of two different maximal ideals of Euclidean ring contain prime element?

Question

Can intersection of two different maximal ideals of Euclidean ring contain prime element?

We define $I$ maximal ideal of ring $R$, if there is no such ideal $I’ \neq R$, that $I \subset I’ \subset R$.

Answer 1

Let $p$ be a prime element in the ring.

Then $(p)$ is maximal as if $(p)\subset (m)$, then $m\mid p$ which implies that either $m$ is a unit (in which case $(m) = (1)$) or $(m) = (p)$, either case, we see that $(p)$ is the largest pricipal proper-ideal in the ring.

Now, Euclidean rings are PIDs, so, $(p)$ is maximal.