  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 $\scal(R(T))$.
