Editing Valuation ring (section)

{{Short description|Concept in algebra}}
In [[abstract algebra]], a '''valuation ring''' is an [[integral domain]] ''D'' such that for every non-zero element ''x'' of its [[field of fractions]] ''F'', at least one of ''x'' or ''x''<sup>&minus;1</sup> belongs to ''D''.

Given a [[field (mathematics)|field]] ''F'', if ''D'' is a [[subring]] of ''F'' such that either ''x'' or ''x''<sup>&minus;1</sup> belongs to
''D'' for every nonzero ''x'' in ''F'', then ''D'' is said to be '''a valuation ring for the field ''F''''' or a '''place''' of ''F''. Since ''F'' in this case is indeed the field of fractions of ''D'', a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field ''F'' is that valuation rings ''D'' of ''F'' have ''F'' as their field of fractions, and their [[ideal (ring theory)|ideals]] are [[totally ordered]] by [[subset|inclusion]]; or equivalently their [[principal ideal]]s are totally ordered by inclusion. In particular, every valuation ring is a [[local ring]].

The valuation rings of a field are the maximal elements of the set of the local subrings in the field [[partial order|partially ordered]] by '''dominance''' or '''refinement''',{{sfn|Hartshorne|1977|loc=Theorem I.6.1A}} where
:<math>(A,\mathfrak{m}_A)</math> dominates <math>(B,\mathfrak{m}_B)</math> if <math>A \supseteq B</math> and <math>\mathfrak{m}_A \cap B = \mathfrak{m}_B</math>.{{sfn|Efrat|2006|p=55}}

Every local ring in a field ''K'' is dominated by some valuation ring of ''K''.

An integral domain whose [[localization of a ring|localization]] at any [[prime ideal]] is a valuation ring is called a [[Prüfer domain]].