Real closed ring

In mathematics, a real closed ring is a commutative ring "A" that is a subring of a product of real closed fields, which is closed under continuous semi-algebraic functions defined over the integers.
Examples of real closed rings.
Since the rigorous definition of a real closed ring is of technical nature it is convenient to see a list of prominent examples first. The following rings are all real closed rings: 
Definition.
A real closed ring is a reduced, commutative unital ring "A" which has the following properties:
The link to the definition at the beginning of this article is given in the section on algebraic properties below.
The real closure of a commutative ring.
Every commutative unital ring "R" has a so-called real closure rcl("R") and this is unique up to a unique ring homomorphism over "R". This means that rcl("R") is a real closed ring and there is a (not necessarily injective) ring homomorphism formula_2 such that for every ring homomorphism formula_3 to some other real closed ring "A", there is a unique ring homomorphism formula_4 with formula_5.
For example, the real closure of the polynomial ring formula_6 is the ring of continuous semi-algebraic functions formula_7.
An arbitrary ring "R" is semi-real (i.e. −1 is not a sum of squares in "R") if and only if the real closure of "R" is not the null ring.
The real closure of an ordered field is in general not the real closure of the underlying field. For example, the real closure of the ordered subfield formula_8 
of formula_9 is the field formula_10 of real algebraic numbers, whereas the real closure of the field formula_8 is the ring formula_12 (corresponding to the two orders of formula_8). More generally the real closure of a field "F" is a certain subdirect product of the real closures of the ordered fields ("F","P"), where "P" runs through the orderings of "F".
Model theoretic properties.
The class of real closed rings is first-order axiomatizable and undecidable. The class of all real closed valuation rings is decidable (by Cherlin-Dickmann) and the class of all real closed fields is decidable (by Tarski). After naming a definable radical relation, real closed rings have a model companion, namely von Neumann regular real closed rings.
Comparison with characterizations of real closed fields.
There are many different characterizations of real closed fields. For example,
in terms of maximality (with respect to algebraic extensions): a real closed field is a maximally orderable field; or, a real closed field (together with its unique ordering) is a maximally ordered field. Another characterization says that the intermediate value theorem holds for all polynomials in one variable over the (ordered) field. In the case of commutative rings, all these properties can be (and are) analyzed in the literature. They all lead to different classes of rings which are unfortunately also called "real closed" (because a certain characterization of real closed fields has been extended to rings). None of them lead to the class of real closed rings and none of them allow a satisfactory notion of a closure operation. A central point in the definition of real closed rings is the globalisation of the notion of a real closed field to rings when these rings are represented as rings of functions on some space (typically, the real spectrum of the ring).