Is the ring $\mathbb{Z}_{p^n}$ local?

Question

Let $p$ be a prime number. Is the ring $\mathbb Z_{p^n}$ a local ring? That is, the set of non-units an ideal of the ring?

I think yes, because the only prime that divides the order of the ring is $p$, so the number of maximal ideals is also $1$. Any hints? Thanks beforehand

Answer 1

Another way:

If $R$ is a commutative ring and $M$ is a maximal ideal of $R$, then $R/M^k$ is a local ring.

Hint: The maximal ideals of $R/M^k$ correspond with the maximal ideals of $R$ containing $M^k$. Suppose $P$ is such an ideal (keeping in mind that maximal ideals are prime!) What does $M^k\subseteq P$ imply?