Terminal spaces of monoids
Amartya Goswami
TL;DR
The paper develops a Zariski-type topology on the set $\mathcal{S}(M)$ of strongly irreducible ideals of a commutative monoid $M$, extending analogous results from rings, semirings, and modules. It shows that the hull-kernel operator $\mathcal{HK}$ yields a closed-set topology, making $\mathcal{S}(M)$ a quasi-compact $T_0$ space, with each nonempty irreducible closed set having a unique generic point. Irreducible closed subsets correspond to strongly irreducible ideals via $\mathcal{K}(X)$, and irreducible components correspond to minimal strongly irreducible ideals; density of $\mathrm{Max}(M)$ and $\mathrm{Spec}(M)$ in $\mathcal{S}(M)$ is characterized by equalities among $m$-, $p$-, and $s$-radicals. The results characterize arithmetic monoids topologically and discuss Noetherian cases and invertibility criteria within the terminal space framework.
Abstract
The purpose of this note is a wide generalization of the topological results of various classes of ideals of rings, semirings, and modules, endowed with Zariski topologies, to strongly irreducible ideals (endowed with Zariski topologies) of monoids, called terminal spaces. We show that terminal spaces are $T_0$, quasi-compact, and every nonempty irreducible closed subset has a unique generic point. We characterize arithmetic monoids in terms of terminal spaces. Finally, we provide necessary and sufficient conditions for the subspaces of maximal and prime ideals to be dense in the corresponding terminal spaces.