Rings of formal power series is finitely generated as module.

Question

The question rises when I was reading Kemper's "A course in Commutative Algebra." Let $K$ be a field and consider the ring of formal power series $K[[x]]$. In an exercise I proved $K[[x]]$ is not Jacobson, hence not finitely generated as a K-algebra. But since $K$ is Noetherian and $K[[x]]$ is a K-module, $K[[x]]$ is Noetherian is equivalent to $K[[x]]$ is finitely generated as a K-module. There is a proof of $K[[x]]$ being Noetherian. I can't see why $K[[x]]$ is a finitely generated K-module but not a finitely generated K-algebra. Is there a mistake in my understanding? Any help is appreciated.

Answer 1

$K[[x]]$ is a Noetherian ring, not a Noetherian $K$-module. In other words, every ideal in the ring $K[[x]]$ is finitely generated, rather than every $K$-submodule of $K[[x]]$ being finitely generated.