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Topological Properties of Semigroup Primes of a Commutative Ring by Finocchiaro, Carmelo A.; Fontana, Marco; Spirito, Dario is a scholarly article available to read on EtoBox.

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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

Author

Finocchiaro, Carmelo A.; Fontana, Marco; Spirito, Dario

Published

2017

Language

EN