Example 2

Let $K$ be a commutative ring, $A$ the ring of formal power series

(Algebra, Chapter IV, § 5) and $m$ the ideal of $A$ whose elements are the formal power series with no constant term. Let us give $A$ the $m$-adic filtration (no. 1, Example 3); if $M_1,\ldots,M_s$ are the distinct monomials in $X_1,\ldots,X_r$ of total degree $n-1$, clearly every formal power series $u$ of total order $\omega(u)\geq n$

(loc. cit., no. 2) may be written as $\displaystyle\sum_{k=1}^{s}u_kM_k$, where the $u_k$ belong to $m$; it is seen that $m^n$ is the set of formal power series $u$ such that $\omega(u)\geq n$, which shows that $\omega$ is the orderfunction for the $m$-adic filtration. Then clearly, for every formal power series $u\in m^n$, there exists a unique homogeneous polynomial of degree $n$ in the $X_i$ which is congruent to $u$ mod. $m^{n+1}$, namely the sum of terms of degree $n$ of $u$; we conclude that $\operatorname{gr}(A)$ is canonically isomorphic to the polynomial ring $K[X_1,\ldots,X_r]$.