Rings with involution and orderings (Q1818399)

The aim of the paper is to generalize the Artin-Schreier theory of ordered fields to rings with involution. The notion of ordering of a ring with involution is studied, and is related to the formation of rings of fractions in which symmetric elements are invertible. Remarks: Some of the proofs contain gaps or errors. For example, in the proof of Theorem (17) the author claims (implicitly) that if \(M\) is a multiplicatively closed semi-ordering and \(s=s^*\notin M\) then \(M\cup-sP\cup M-sP\) is a multiplicatively closed semi-ordering. This is not true.