Lattice-ordered rings and function rings

Abstract

Introduction: This paper treats the structure of those lattice-ordered rings which are subdirect sums of totally ordered rings—the f-rings of Birkhoff and Pierce [4]. Broadly, it splits into two parts, concerned respectively with identical equations and with ideal structure; but there is an important overlap at the beginning. D. G. Johnson has shown [9] that not every /-ring is unitable, i.e. embeddable in an/-ring which has a multiplicative unit; and he has given a characterization of unitable /-rings. We find that they form an equationally definable class. Consequently in each /-ring there is a definite Z-ideal which is the obstruction to embedding in an /-ring with unit. From Johnson's results it follows that such an ideal must be nil; we find it is nilpotent of index 2, and generated by left and right annihilators. Tarski has shown [13] that all real-closed fields are arithmetically equivalent. It follows easily that every ordered field satisfies all ringlattice identities valid in the reals (or even in the rationals); and from a theorem of Birkhoff [2], every ordered field is therefore a homomorphic image of a latticeordered ring of real-valued functions. Adding results of Pierce [12] and Johnson [9], one gets the same conclusion for commutative /-rings which have no nonzero nilpotents. We extend the result to all zero /-rings, and all archimedean /-rings. We call these homomorphic images of /-rings of real functions formally real /-rings. Birkhoff and Pierce showed [4] that /-rings themselves form an equationally definable class of abstract algebras, defined by rather simple identities involving no more than three variables. The same is true for unitable/-rings. However, no list of identities involving eight or fewer variables characterizes the formally real /-rings. The conjecture is that 'eight can be replaced by any n, but we cannot prove this. We call an element e of an /-ring a superunίt if ex ^ x and xe ^ x for all positive x; we call an /-ring infinitesimal if x2 ^ \x identically. A totally ordered ring is unitable if and only if it has a superunit or is infinitesimal. A general unitable /-ring is a subdirect sum of two summands, L, I, where L is a subdirect sum of totally ordered rings having superunits (we say L has local superunits) and /is infinitesimal. The summand L is unique. We call a maximal J-ideal M in an /-ring A supermodular if AjM has a superunit. The supermodular maximal Z-ideals of A, in the hullkernel topology, form a locally compact Hausdorff space ^{A). If A