Order theoretic and topological Characterizations of the Divided Spectrum of a Ring

Abstract

Let $R$ be a commutative ring with identity. We denote by $\mathcal{D}\mathrm{iv}(R)$ the divided spectrum of $R$ (the set of all divided prime ideals of $R$). By a divspectral space, we mean a topological space homeomorphic with the subspace $\mathcal{D}\mathrm{iv}(R)$ of $\mathrm{Spec}(R)$ endowed with the Zariski topology, for some ring $R$. A divspectral set is a poset which is order isomorphic to $(\mathcal{D}\mathrm{iv}(R),\subseteq)$, for some ring $R$. The main purpose of this paper is to provide some topological (resp., algebraic) characterizations of of divspectral spaces (resp., sets).