Corollary 6

For every commutative Noetherian ring C and every integer n > 0, the ring of formal power series $\mathbf{C}[[X_1, \ldots, X_n]]$ is Noetherian.

This follows from Corollary 5 and no. 6, Corollary to Proposition 6, for if m is the ideal of $A = \mathbf{C}[[X_1, \ldots, X_n]]$ consisting of the formal power series with no constant term, $A/m$ is isomorphic to C and $m/m^2$ to the C-module $\mathbf{C}"$.

Remarks

(1) Corollaries 2, 3 and 6 apply in particular if C is a commutative field.

\* (2) Let g be a Lie algebra over a commutative Noetherian ring C and suppose that g is a finitely generated C-module. Let the enveloping algebra U of g be given the increasing filtration $(U_n)$ defined in no. 3, Example 4. With the corresponding topology, U is discrete and hence Hausdorff and complete; the associated graded ring gr(U) is a finitely generated C-algebra, being a quotient of the symmetric algebra $\mathscr{S}(g)$, hence gr(U) is a Noetherian ring (Corollary 3) and we deduce that U is a left and right Noetherian ring (no. 9, Corollary 2 to Proposition 12). \*