Topological properties of semigroup primes of a commutative ring

A semigroup prime of a commutative ring R is a prime ideal of the semigroup (R, ·). One of the purposes of this paper is to study, from a topological point of view, the space S(R) of prime semigroups of R. We show that, under a natural topology introduced by B. Olberding in 2010, S(R) is a spectral space (after Hochster), spectral extension of Spec(R) , and that the assignment R↦ S(R) induces a contravariant functor. We then relate—in the case R is an integral domain—the topology on S(R) with the Zariski topology on the set of overrings of R. Furthermore, we investigate the relationship between S(R) and the space X(R) consisting of all nonempty inverse-closed subspaces of Spec(R) , which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that S(R) is a spectral retract of X(R) and we characterize when S(R) is canonically homeomorphic to X(R) , both in general and when Spec(R) is a Noetherian space. In particular, we obtain that, when R is a Bézout domain, S(R) is canonically homeomorphic both to X(R) and to the space Overr(R) of the overrings of R (endowed with the Zariski topology). Finally, we compare the space X(R) with the space S(R(T)) of semigroup primes of the Nagata ring R(T), providing a canonical spectral embedding X(R) ↪ S(R(T)) which makes X(R) a spectral retract of S(R(T)).