maximal ideals of ring of formal power series

Suppose that $R$ is a commutative ring with non-zero unity.

If $𝔪$ is a maximal ideal of $R$, then  $𝔐:=𝔪+(X)$  is a maximal ideal of the ring $R[[X]]$ of formal power series.

Also the converse is true, i.e. if $𝔐$ is a maximal ideal of $R[[X]]$, then there is a maximal ideal$𝔪$ of $R$ such that  $𝔐=𝔪+(X)$.

Note.  In the special case that $R$ is a field, the only maximal ideal of which is the zero ideal $(0)$, this corresponds to the only maximal ideal $(X)$ of $R[[X]]$ (see http://planetmath.org/node/12087formal power series over field).

We here prove the first assertion.  So, $𝔪$ is assumed to be maximal.  Let

$$f(x):=a_0+a_{1}X+a_2{X}^2+\mathrm{\dots }$$

be any formal power series in $R[[X]]\setminus 𝔐$.  Hence, the constant term $a_0$ cannot lie in $𝔪$.  According to the criterion for maximal ideal, there is an element $r$ of $R$ such that  $1+r{a}_0\in 𝔪$.  Therefore

$$1+rf(X)=(1+r{a}_0)+r(a_1+a_{2}X+a_3{X}^2+\mathrm{\dots })X\in 𝔪+(X)=𝔐,$$

whence the same criterion says that $𝔐$ is a maximal ideal of $R[[X]]$.