Generalized orderings and rings of fractions (Q2491962)

For a commutative domain every total order can be extended (uniquely) to its field of fractions. This result is generalized to non-commutative domains, weakening the notion of total order in the following sense. Let \(R\) be an associative ring with identity and let \(G\) be a group containing an element \(a\in G\) of order two. A subset \(P\subseteq R\) is called a \(G\)-ordering of \(R\) if there exists a surjetive mapping \(f:R\to G_0\) (the group \(G\) with a zero adjoined) such that: \(f(a)=a\), for any \(x,y \in R\) \(f(xy)=f(x)f(y)\), \(f(x+y)= f(y)\) if either \(f(x)=0\) or \(f(x)=f(y)\), and \(f^{-1}(\{0,1\})=P\). \(G\)-orderings generalize the notion of orderings of higher level on commutative rings studied by \textit{R. Berr} [Manuscr. Math. 75, No. 3, 273--277 (1992; Zbl 0758.12003)]. This generalization is worked out considering the following types of rings of fractions: Ore extensions, maximal (right or two-sided) rings of fractions, division hulls of free algebras and epic fields.