$A[[X]]$ is a regular ring. power series over regular ring is regular Assume $A$ is a com. ring with unity.
I am trying to find a more elementary proof on the following statement, than what I have read so far (see the list at the end).
If $A$ is a regular ring, so is $A[[X]]$.
This is what I have come up so far:
Suppose $\mathfrak{M}$ be a maximal ideal of $R = A[[X]]$. Then $\mathfrak{M}$ is of the form $R\mathfrak{m}+ RX$ with $\mathfrak{m}$ being a maximal ideal of $A$ (the proof for this statement will be added).
Say $x_1,\ldots,x_n$ are the generators of $\mathfrak{m}$ in $A$ with minimal $n$.
Since $A$ is regular $ht(\mathfrak{m}A_\mathfrak{m}) =n$.
The dimension of the local ring $R_\mathfrak{M}$ equals the height of its maximal ideal, which is
\begin{align}{
ht(\mathfrak{M}R_\mathfrak{M})=ht((x_1,\ldots,x_n,X))}.
\end{align}
To finish this proof, one has to show $ht((x_1,\ldots,x_1,X))=ht(\mathfrak{m})+1$ (the inequality $ht((x_1,\ldots,x_1,X)) \geq ht(\mathfrak{m})+1$ seems trivial). Then $ht(\mathfrak{m})+1 = n+1$ and I believe it's done.
However, I am not yet able to do it.
Could theorem 15.1 in Commutative Ring Theory by Matsumura help with the problem stated?
The approaches I have seen so far:
ON UNIQUE FACTORIZATION DOMAINS
BY PIERRE SAMUEL, Some Remarks on Factorization in Power Series Rings
DAVID A. BUCHSBAUM, Commutative Ring Theory by Matsumura, TOPICS IN m-ADICTOPOLOGIES by s.Greco and P. Salmon.
 A: Notice that $R/\mathfrak{m}R\cong k[[X]]$ where $k=A/\mathfrak{m}$. Now if $S=R\setminus \mathfrak{M}$ so that $R_{\mathfrak{M}}=S^{-1}R$, and $\bar{S}$ is the image of $S$ in $R/\mathfrak{m}R$, notice that we have the canonical isomorphism
$$
R_{\mathfrak{M}}/\mathfrak{m}R_{\mathfrak{M}}\cong \bar{S}^{-1}(R/\mathfrak{m}R).
$$
Now as $\mathfrak{M}$ contains $X$, the image of $\bar{S}$ in $k[[X]]$ lies in the complement of $(X)$. But $k[[X]]$ is local with maximal ideal $(X)$ (every power series with a constant term has a mulhttiplicative inverse, which can easily be constructed by induction), and thus every element of $\bar{S}$ is a unit. Hence
$$
R_{\mathfrak{M}}/\mathfrak{m}R_{\mathfrak{M}}\cong \bar{S}^{-1}(R/\mathfrak{m}R)\cong R/\mathfrak{m}R\cong k[[X]].
$$
Finally, $k[[X]]$ is a DVR with valuation $v\left(\sum_i a_iX^i\right):=\inf\{i\ \mid\ a_i\neq 0\}$ (this is again due to the fact that you can invert power series with non-zero constant term). So $\operatorname{ht}((X))=1$, because the only prime ideals of $k[[X]]$ are $(0)$ and $(X)$.
