Ring of formal power series is not always local ring

Question

Let $R$ be a local ring,then,I know $R[[x]]$ is also local ring.
But what about when $R$ is not local ring?

For example, I heard $\mathbb Z[a1,a2,a3,a4,a5,a6][[x]] $($a1,a2,a3,a4,a5,a6∈K$:field)
is not local ring.Why this is not local ring?

Answer 1

You can even take, for instance, $\mathbb{Z}[[X]]$. The ring $\mathbb{Z}$ is not not local, since all ideals of the form $p\mathbb{Z}$ with $p$ a prime number, are maximal. Then, $\mathfrak{m} := p\mathbb{Z} +(X) = p\mathbb{Z}+ X\cdot\mathbb{Z}[[X]]$ is a maximal ideal in $\mathbb{Z}[[X]]$. Therefore, $\mathbb{Z}[[X]]$ is not local, since it has no unique maximal ideal.