We present a constructive interpretation of some results in [6]. To any integral domain R we associate a distributive lattice V (R) which is a point-free presentation of the space of valuations over the ring R. Our definition, by generators and relations, is similar to Joyal’s definition of the Zariski lattice of a ring R, which is a point-free presentation of the Zariski spectrum of R. The spa...

All rings in this paper are commutative with unity; we will deal mainly with integral domains. Let R be a ring with total quotient ring K. A fractional ideal I of R is invertible if II−1 = R; equivalently, I is a projective module of rank 1 (see, e.g., [Eis95, Section 11.3]). Here, I−1 = (R : I) = {x ∈ K |xI ⊆ R}. Moreover, a projective R-module of rank 1 is isomorphic to an invertible ideal. (...

Given a significative class F of commutative rings, we study the precise conditions under which a commutative ring R has an F -envelope. A full answer is obtained when F is the class of fields, semisimple commutative rings or integral domains. When F is the class of Noetherian rings, we give a full answer when the Krull dimension of R is zero and when the envelope is required to be epimorphic. ...

We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula—plus a few other technical requirements. The theory of vector spaces and the theory fields are examples. We prove the amalgamation property and the existence of a model-companion. We show that the model-com...

Proposition 1.1. (a) Any open immersion is étale. (b) The composite of two étale morphisms is étale. (c) Any base change of an étale morphism is étale. (d) If φ ◦ ψ and φ are étale, then so is ψ. Proposition 1.2. Let f : X → Y be an étale morphism. (a) For all x ∈ X, OX,x and OY,f(x) have the same Krull dimension. (b) The morphism f is quasi-finite. (c) The morphism f is open. (d) If Y is reduc...

We continue the study of a lattice-ordered ring G(X), associated with the ring C(X). Following [10], X called RG when G(X) = C(Xδ). An RG-space must have a dense set of very weak P-points. It must have a dense set of almost-P-points if Xδ is Lindelöf, or if the continuum hypothesis holds and C(X) has small cardinality. Space which are RG must have finite Krull dimension when taken with respect ...

Let k be a field. We show that a finitely generated simple Goldie k-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and l...

Remark 1.6. In [Mor06], this result is stated without proof. Our goal here is to indicate that the result follows without any of the machinations appearing in §3.2 of [Mor06] (where the result is stated). Ayoub has shown that if the Krull dimension of k is ≥ 2, then the unstable A-connectivity property will fail, thus the strong A-invariance property cannot hold over 2-dimensional rings. Howeve...