No abstract

No abstract

Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result:Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that:(1) The symbol “∣” is interpreted as the honest divisibility relation: (2) The following divisibility property holds in T:If T admits q.e. in ℒ, then T = RCVR.We do not know at present whether the restriction imposed by condition (2) can be weakened.The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings, for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings:Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent:(a) A is a convexly ordered valuation ring.(b) Every ideal (or, equivalently, principal ideal) is convex in A.(c) A is a valuation ring convex in its field of fractions quot(A).(d) A is a valuation ring and its maximal ideal MA is convex (in A or, equivalently, in quot (A)).(e) A is a valuation ring and its maximal ideal is bounded by ± 1.

No abstract

No abstract

Following the introduction of an algebraic K-theory of special groups in [Dickmann and Miraglia, Algebra Colloq. 10 (2003) 149-176], generalizing Milnor's mod 2 K-theory for fields, the aim of this paper is to compute the K-theory of Boolean algebras, inductive limits, finite products, extensions, SG-sums and (finitely) filtered Boolean powers of special groups. A parallel theme is the preservation by these constructions of property [SMC], an analog for the K-theory of special groups of the property "multiplication by l(−1) is injective" in Milnor's mod 2 K-theory (see [Milnor, Invent. Math. 9 (1970) 318-344]).

To each pair, R, T , consisting of a unitary commutative von Neumann-regular ring, R, where 2 is a unit and T is a preorder on R, we associate a reduced special group, G T (R), which faithfully reflects quadratic form theory, modulo T , over free R-modules and then show, using the representation of R as the ring of global sections of its affine scheme, together with results from [M. Dickmann, F. Miraglia, On quadratic forms whose total signature is zero mod 2 n . Solution to a problem of M. Marshall, Invent. Math. 133 (1998) 243-278; M. Dickmann, F. Miraglia, Lam's Conjecture, Algebra Colloq. 10 (2003) 149-176; M. Dickmann, F. Miraglia, Algebraic K-theory of special groups, J. Pure Appl. Algebra 204 (2006) 195-234], that G T (R) satisfies a powerful K-theoretic property, the [SMC]-property. From this we conclude that quadratic form theory modulo T over free R-modules verifies Marshall's signature conjecture, Lam's conjecture, as well as a reduced version of Milnor's Witt ring conjecture.The main result of this paper is that, if R is a commutative von Neumann-regular ring in which 2 is a unit, then the reduced theory of quadratic forms with invertible coefficients in R,

No abstract