I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the form $h(z) = \frac{P(z)}{Q(z)}$ with $P$, $Q$ polynomials in $\mathbb{C}$ with integer coefficients and for even $p_i\in\mathbb{N}$ and some $n\in\mathbb{N}$ $$Q(z)=\prod_{i=1}^n (z^{p_i}-1),$$ and $deg(Q)> deg(P)$. Then, $S$ together with the usual addition and multiplication forms a ring, commutative without unity, because the constant functions are not included in $S$.
I now consider the scheme $\mathfrak{S}$ over $S$ for which I need to compute the prime ideals of $S$, to obtain the spectrum of $S$. Here, I am already struggeling since my knowledge on polynomial rings does not seem to transfer to $S$. Could someone give me a pointer on the prime ideals?
For the final application, in particular, given a sequence $(h_k)_{k\in\mathbb{N}}$ with $Q_{k}|Q_{k+1}$ and $deg(Q_k)<deg(Q_{k+1})$, I look at the quotient rings $\mathcal{S}_q:=\{S/\langle h_1,...,h_k\rangle|k\in\mathbb{N}\}$. Each $\tilde{S}_k\in\mathcal{S}_q$ is itself a commutative ring and as such one can associate a scheme $\mathfrak{S}_k$ to it. I am wondering how those schemes behave with growing $k$ relativ to the initial one $\mathfrak{S}$. Is there any literature on these types of questions?
