Noncommutative real algebraic geometry studies the real spectrum (i.e. the set of all orderings) of an associative ring and its generalizations (e. g. the set of all orderings of higher level, the set of all *-orderings). Interesting rings with nonempty real spectrum (i.e. formally real rings) are (large classes of) quantum groups and rings of differential operators. For some rings, it is possible to give an explicit description of its real spectrum by classifying all its orderings. The basic algebraic results of (commutative) real algebraic geometry can usually be extended to the noncommutative setting. Unfortunately, the Lang homomorphism theorem fails for most noncommutative rings and a good substitute for it is still being sought for.
If the real spectrum of a ring is empty, then -1 is a sum of permuted products of squares. The length of the shortest such sum is called the product level of the ring. The product level can be viewed as an obstruction to orderability. Usually, it is very difficult to compute. Sometimes it can be estimated by other invariants that are easier to compute. Full matrix rings of dimension at least 2 have small product levels (1 or 2) and are considered as being the most distant from orderable rings. Product levels of skew fields are connected to Witt rings of skew fields.
Noncommutative valuation rings in the sense of Schilling come naturally as sets of bounded elements of ordered skew fields. Classification of all real valuations on a given skew field D is the essential part of the classification of all orderings of D. Matrix valuations appear naturally in the study of free fields.