Given a (pointed or unpointed) monoid $M$, one defines an ideal of $M$ to be a subset $I$ of $M$ such that if $a\in I$ and $r\in M$, then $ra\in I$ (and such that $0\in I$ if $M$ is pointed). A prime ideal $\frak p$ of $M$ is then an ideal such that if $ab\in\frak p$, then $a\in\frak p$ or $b\in\frak p$. The set of all prime ideals of $M$ can is written $\mathrm{MSpec}(M)$, and it can be made into a topological space equipped with a sheaf of monoids, giving a variant of algebraic geometry where monoids play the role of rings. Good references for this are Sections 1--2 of this paper and Ogus's book.

Lately I've been trying to compile some examples of such prime spectra, and wondering about how these relate to the usual $\mathrm{Spec}$ of a ring. So far, I've found or read about the following examples (the first five examples below come from Martin Brandenburg's answer here):

  • $\mathrm{MSpec}(\mathbf{N},+)=\{0,\mathbf{N}_{>0}\}$.
  • $\mathrm{MSpec}(\mathbf{Z},+)=\{0\}$.
  • $\mathrm{MSpec}(\mathbf{N},\cdot)=\mathrm{MSpec}(\mathbf{Z},\cdot)=\mathcal{P}(\mathbf{P})$, the powerset of the set $\mathbf{P}$ of all prime numbers.
  • $\mathrm{MSpec}(\mathbf{N}\otimes_{\mathbf{N}_+}\mathbf{N})\cong\mathcal{P}(\mathbf{P}\times\mathbf{P})$.
  • $\mathrm{MSpec}(\mathbf{Z}\otimes_{\mathbf{F}_1}\mathbf{Z})\cong\mathcal{P}(\mathbf{P})\times\mathcal{P}(\mathbf{P})$.
  • $\mathrm{MSpec}(K)=\{(0)\}$ whenever $K^\times=K\setminus\{0\}$. In particular this applies to $\mathbf{F}_1=\{0,1\}$.
  • $\mathrm{MSpec}(\mathbf{F}_1[x])=\{(0),(x)\}$.
  • $\mathrm{MSpec}(\mathbf{F}_1[x,y])=\{(0),(x),(y),(x,y)\}$.
  • $\mathrm{MSpec}(\mathbf{F}_1[x_1,...,x_n])=\mathcal{P}(\{x_1,...x_n\})$.
  • $\mathrm{MSpec}(\mathbf{F}_1[t,t^{-1}])=\{(0),(t),(t^{-1})\}$.

Now, any ring $R$ has an associated monoid, given by keeping only multiplication, and these should give more examples. For the usual prime spectrum of a ring, we have \begin{align*} \mathrm{Spec}(\mathbf{C}[x]) &= (0)\cup\bigcup_{a\in\mathbf{C}}(x-a),\\ \mathrm{Spec}(\mathbf{C}[x,y]) &= (0)\cup\bigcup_{a,b\in\mathbf{C}}(x-a,y-b)\cup\{p(x,y):\text{$p\in\mathbf{C}[x,y]$ irreducible}\}\\ \mathrm{Spec}(\mathbf{Z}_p) &= (0)\cup(p) \end{align*} and $\mathrm{Spec}(\mathbf{Z}[x])$ is known and described in Spectrum of $\mathbb{Z}[x]$. Do we know what are the $\mathrm{MSpec}$ of the underlying pointed monoids of these four rings?



You must log in to answer this question.