Topological properties of semigroup primes of a commutative ring

Abstract

A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space \scal(R) of prime semigroups of $R$. We show that, under a natural topology introduced by B. Olberding in 2010, \scal(R) is a spectral space (after Hochster), spectral extension of \Spec(R), and that the assignment R\mapsto\scal(R) induces a contravariant functor. We then relate -- in the case $R$ is an integral domain -- the topology on \scal(R) with the Zariski topology on the set of overrings of $R$. Furthermore, we investigate the relationship between \scal(R) and the space $\boldsymbol{\mathcal{X}}(R)$ consisting of all nonempty inverse-closed subspaces of \spec(R), which has been introduced and studied in C.A. Finocchiaro, M. Fontana and D. Spirito, "The space of inverse-closed subsets of a spectral space is spectral" (submitted). In this context, we show that \scal( R) is a spectral retract of $\boldsymbol{\mathcal{X}}(R)$ and we characterize when \scal( R) is canonically homeomorphic to $\boldsymbol{\mathcal{X}}(R)$, both in general and when \spec(R) is a Noetherian space. In particular, we obtain that, when $R$ is a B\'ezout domain, \scal( R) is canonically homeomorphic both to $\boldsymbol{\mathcal{X}}(R)$ and to the space \overr(R) of the overrings of $R$ (endowed with the Zariski topology). Finally, we compare the space $\boldsymbol{\mathcal{X}}(R)$ with the space \scal(R(T)) of semigroup primes of the Nagata ring $R(T)$, providing a canonical spectral embedding \xcal(R)\hookrightarrow\scal(R(T)) which makes \xcal(R) a spectral retract of \scal(R(T)).Comment: 21 page