A question about the ideals of a ring of power series.

Question

Let $F$ a field and we consider the ring of power series $F[[x]]$. After proving that $F[[x]]$ is a P.I.D. and after proving that the ideals of $F[[x]]$ are of the form $(x^k)$ for $k\ge0$, why $F[x]\supset (x)\supset(x^2)$? Why $F[[x]]\ne (x)$ and $(x)\ne (x^2)$.

Thanks!

Answer 1

$(x)$ is the ideal consisting of series with no constant term. Certainly there exist series with nonzero constant term, hence the inclusion is proper. Similarly, $(x^2)$ is the ideal of series with no constant or linear term. Since there exist series with constant terms, or with no constant term but a linear term, the inclusion is proper.