Uniqueness of maximal ideals in factor rings of formal power series

Question

I need to show the uniqueness of a maximal ideal in the ring $S_{n}=\mathbb{C}[[z]]/z^{n}\mathbb{C}[[z]]$. The ideal in question is the ideal $I/I^{n}$ where $I^{n}=z^{n}\mathbb{C}[[z]]$. Now I know that $S_{n} \cong \mathbb{C}[z]/(z^n)$ as rings so I guess the question can be asked their in analogue and I know that $S_{n}$ is a PID. However I can't find a good description of the ideal $I/I^{n}$ or whether it is even an ideal. To prove it is maximal I guess I can determine this by finding its generator when considered in $\mathbb{C}[z]/(z^n)$ and prove it is irreducible as a polynomial, but how do I go about proving uniqueness?

Answer 1

The most important thing to know about the ring of power series over a field ($F[[z]]$) is that its nonzero ideals are exactly of the form $(z^n)$ (counting the whole ring as $(z^0)$.) Consequently its ideals are all linearly ordered, and therefore it has a unique maximal ideal.

By the correspondence theorem for quotient rings, every possible quotient of a ring with a unique maximal ideal has a unique maximal ideal (the image of the original maximal ideal.)

So you see it is not all that important that the ideals are principal or even finitely generated since this is true of all local rings with unique maximal ideals (they are called local rings.)

The method you've chosen will work, too. In a PID, the ideals containing $(x)$ are principal ideals generated by divisors of x, and hopefully you know the divisors of $z^n$. By correspondence, again, your quotient is a ring with linearly ordered ideals, and so there is a unique maximal ideal.

and I know that $S_{n}$ is a PID

Er, it's not a PID since z is usually a zero divisor in it. But yeah, it is at least a principal ideal ring.