A nice introduction to commutative real algebra is the book of Knebusch & Scheiderer:
- KNEBUSCH, Manfred; SCHEIDERER, Claus;
Einfhrung in die reelle Algebra. (German) [Introduction to real algebra]
Vieweg Studium: Aufbaukurs Mathematik [Vieweg Studies: Mathematics
Course], 63.Friedr. Vieweg & Sohn, Braunschweig, 1989.
First steps in commutative real algebraic geometry are done in Lam's
- LAM, T. Y. An introduction to real algebra.
Ordered fields and real algebraic geometry (Boulder, Colo., 1983).
Rocky Mountain J. Math. 14 (1984), no. 4, 767--814.
For some deeper results in this direction, the reader can consult
- BOCHNAK, J.; COSTE, M.; ROY, M.-F.
Géométrie algébrique réelle. (French) [Real algebraic geometry]
Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in
Mathematics and Related Areas (3)], 12.
Springer-Verlag, Berlin, 1987.
Noncommutative real algebra was developed in a series of papers by
- LEUNG, Ka Hin; MARSHALL, Murray; ZHANG, Yufei;
The real spectrum of a noncommutative ring.
J. Algebra 198 (1997), no. 2, 412--427.
- MARSHALL, Murray; ZHANG, Yufei;
Orderings, real places, and valuations on noncommutative integral domains.
J. Algebra 212 (1999), no. 1, 190--207.
- MARSHALL, Murray; ZHANG, Yang;
Orderings and valuations on twisted polynomial rings. (English. English
Comm. Algebra 28 (2000), no. 8, 3763--3776.
In connection with this one should also mention the theory of orderings of
It was started by Becker in the seventies, continued by Berr, Craven,
In the last decade of the previous century a connection between
and real algebra was discovered. This has been an active area ever since.
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