The order structure of the real spectrum of commutative rings (Q1577626)

The real spectrum of a commutative ring \(A\) with unit is the set of all pairs \((P,\leq)\) where \(P\) is a prime ideal of \(A\) and \(\leq\) is a total (ring)-order on \(A/P\) endowed with a basis of open sets \(H(a_1,\dots,a_n)\) consisting of all elements \((P,\leq)\) such that \(0<(a_1/P) \wedge\dots \wedge 0< (a_n/P)\), for arbitrary finite sequences of \(A\). The present paper demonstrates that any jump-dense and complete root system is order-isomorphic to the real spectrum of a commutative unitary ring as a significant contribution to the problem of providing an ``intrinsic'' characterization of those special spaces that occur as real spectra of rings. As a byproduct the authors produce from their constructions needed in the proofs of their results several classes of rings with sufficiently nice properties generally of a formal-power-series like form over the real numbers or an inverse limit of such rings over a suitably directed set which may be of interest to students who otherwise might eschew perusing this quite interesting paper.