Introduction. The concept of a local ring was introduced by Krull [7](1), who defined such a ring as a commutative ring 9î in which every ideal has a finite basis and in which the set m of all non-units is an ideal, necessarily maximal. He proved that the intersection of all the powers of m is the zero ideal. If the powers of m are introduced as a system of neighborhoods of zero, then 3Î thus b...

 ‎In this article‎, ‎we first‎ ‎show that non-Noetherian Artinian uniserial modules over‎ ‎commutative rings‎, ‎duo rings‎, ‎finite $R$-algebras and right‎ ‎Noetherian rings are $1$-atomic exactly like $Bbb Z_{p^{infty}}$‎. ‎Consequently‎, ‎we show that if $R$ is a right duo (or‎, ‎a right‎ ‎Noetherian) ring‎, ‎then the Noetherian dimension of an Artinian‎ ‎module with homogeneous uniserial dim...

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...