In mathematics, a real closed ring (RCR) 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.
Since the rigorous definition of a real closed ring is of technical nature it is convenient to see a list of prominent examples first.
For example, the real closure of the polynomial ring
is the ring of continuous semi-algebraic functions
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
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. 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.
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.