Graph components of prime spectra (Q657971)

Every topological space \(X\) carries a relation called \textit{specialization}: Let \(x,y\in X\). Then \(x\) specializes to \(y\) if \(x\in\overline{\{y\}}\). The specialization relation is a partial order if and only if \(X\) is a \(T_0\)-space. Specialization gives the topological space the structure of a directed graph: The vertices are the points of the space; the edges are induced by specialization. The connected components of \(X\) viewed as a topological space are called topological components; the connected components of \(X\) viewed as a graph are called graph components. Every topological component is a union of graph components. An important class of topological spaces are the spectral spaces. By [\textit{M. Hochster}, Trans. Am. Math. Soc. 142, 43--60 (1969; Zbl 0184.29401)] these are exactly the prime spectra of commutative rings with unit. The prime spectrum \(\mathrm{Spec}(A)\) of a commutative ring with unit \(A\) is the set of its prime ideals equipped with the Zariski topology. One may assume that \(A\) is reduced since the prime spectrum doesn't change when going from \(A\) to \(A/\mathrm{nil}(A)\) where \(\mathrm{nil}(A)\) is the ideal of nilpotent elements of \(A\). Specialization in \(\mathrm{Spec}(A)\) coincides with inclusion: Given two prime ideals \(\mathfrak{p}\) and \(\mathfrak{q}\) of \(A\), \(\mathfrak{p}\) specializes to \(\mathfrak{q}\) if and only if \(\mathfrak{p}\subseteq\mathfrak{q}\). In the first main part of the paper the graph components of prime spectra are studied. Interesting results are the following: If a spectral space is normal (i.e. every point specializes to only one maximal point) then its space of maximal points is compact and is homeomorphic to the space of graph components. The space of graph components is Boolean if and only if the topological components is a graph component. In the second main part of the paper the connections between arithmetic properties of a ring and properties of the graph components of its prime spectrum are investigated. The question is if various classes of rings that are defined by properties of the graph components of their prime spectra are elementary classes. The latter means that the class is defined by a list of axioms in the language of rings \(\mathcal{L}=\{0,1,+,-,\cdot\}\). The class of rings with graph connected prime spectrum is for example not elementary. For this an ultraproduct argument is used. Various classes of rings from the literature as von Neumann regular rings, clean rings, or Gel'fand rings can be described by combinatorial properties of its prime spectra viewed as a graph. The paper is well written.