The notions of a preordering and an ordering of a ring R with involution are investigated. An algebraic condition for the existence of an ordering of R is given. Also, a condition for enlarging an ordering of R to an overring is given. As for the case of a field, any preordering of R can be extended to some ordering. Finally, we investigate the class of archimedean ordered rings with involution.
Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel's axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under x ↦ xa² for non-zero a, in place of requiring that positive elements have a positive product. Our aim in this work is to study this type of ordering in the case of a division ring. We show that it actually behaves just as in the commutative...
Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel’s axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under for nonzero , instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative...
The concept of a Prüfer ring is studied in the case of rings with involution such that it coincides with the corresponding notion in the case of commutative rings.
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