Some Applications

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Commutative Ring ◽  
Model Theory ◽  
Galois Theory ◽  
Boundary Operator ◽  
Maximal Spectrum ◽  
Geometric Logic

This chapter describes some applications of the theory developed in the previous chapters in a variety of different mathematical contexts. The main methodology used to generate such applications is the ‘bridge technique’ presented in Chapter 2. The discussed topics include restrictions of Morita equivalences to quotients of the two theories involved, give a solution to a prozblem of Lawvere concerning the boundary operator on subtoposes, establish syntax-semantics ‘bridges’ for quotients of theories of presheaf type, present topos-theoretic interpretations and generalizations of Fraïssé’s theorem in model theory on countably categorical theories and of topological Galois theory, develop a notion of maximal spectrum of a commutative ring with unit and investigate compactness conditions for geometric theories allowing one to identify theories lying in smaller fragments of geometric logic.

On the Galois Theory of Commutative Rings II: Automorphisms induced in Residue Rings

1987 ◽  
Vol 39 (5) ◽  
pp. 1025-1037
Author(s):  
Carl Faith
Keyword(s):  
Commutative Ring ◽  
Galois Theory ◽  
Commutative Rings ◽  
Group Of Automorphisms ◽  
Residue Ring ◽  
Image Position

Let G be a group of automorphisms of a commutative ring K, and let KG denote the Galois subring consisting of all elements left fixed by every g in G. An ideal M is G-stable, or G-invariant, provided that g(x) lies in M for every x in M, that is, g(M) ⊆ M, for every g in G. Then, every g in G induces an automorphism in the residue ring , and if is the group consisting of all , trivially1When the inclusion (1) is strict, then G is said to be cleft at M, or by M, and otherwise G is uncleft at (by) M. When G is cleft at all ideals except 0, then G is cleft, and uncleft otherwise.

scholarly journals The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings

2018 ◽  
Vol 26 (3) ◽  
pp. 41-56
Author(s):  
H. Ansari-Toroghy ◽  
Sh. Habibi
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Zariski Topology ◽  
Topological Properties ◽  
Maximal Spectrum ◽  
Topology Graph

AbstractLet M be a module over a commutative ring and let Max(M) be the collection of all maximal submodules of M. We topologize Max(M) with quasi-Zariski topology, where M is a Max-top module. For a subset T of Max(M), we introduce a new graph $G(\tau_T^{*m})$, called the quasi-Zariski topology-graph on the maximal spectrum of M. It helps us to study algebraic (resp. topological) properties of M (resp. Max(M)) by using the graphs theoretical tools.

On the Maximal Spectrum of Semiprimitive Multiplication Modules

2008 ◽  
Vol 51 (3) ◽  
pp. 439-447
Author(s):  
Karim Samei
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Discrete Space ◽  
Multiplication Module ◽  
Prime Submodules ◽  
Maximal Submodule ◽  
Isolated Points ◽  
Maximal Spectrum ◽  
One Point Compactification

AbstractAnR-moduleMis called a multiplication module if for each submoduleNofM,N=IMfor some idealIofR. As defined for a commutative ringR, anR-moduleMis said to be semiprimitive if the intersection of maximal submodules ofMis zero. The maximal spectra of a semiprimitive multiplication moduleMare studied. The isolated points of Max(M) are characterized algebraically. The relationships among the maximal spectra ofM, Soc(M) and Ass(M) are studied. It is shown that Soc(M) is exactly the set of all elements ofMwhich belongs to every maximal submodule ofMexcept for a finite number. If Max(M) is infinite, Max(M) is a one-point compactification of a discrete space if and only ifMis Gelfand and for some maximal submoduleK, Soc(M) is the intersection of all prime submodules ofMcontained inK. WhenMis a semiprimitive Gelfand module, we prove that every intersection of essential submodules ofMis an essential submodule if and only if Max(M) is an almost discrete space. The set of uniform submodules ofMand the set of minimal submodules ofMcoincide. Ann(Soc(M))Mis a summand submodule ofMif and only if Max(M) is the union of two disjoint open subspacesAandN, whereAis almost discrete andNis dense in itself. In particular, Ann(Soc(M)) = Ann(M) if and only if Max(M) is almost discrete.

scholarly journals Galois Theory for Rings with Finitely Many Idempotents

1966 ◽  
Vol 27 (2) ◽  
pp. 721-731 ◽  
Author(s):  
O. E. Villamayor ◽  
D. Zelinsky
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Fundamental Theorem ◽  
Galois Theory ◽  
Finitely Generated ◽  
Fixed Ring ◽  
Ring Extensions ◽  
A Finite Group

In [5], Chase, Harrison and Rosenberg proved the Fundamental Theorem of Galois Theory for commutative ring extensions S ⊃ R under two hypotheses: (i) 5 (and hence R) has no idempotents except 0 and l; and (ii) 5 is Galois over R with respect to a finite group G—which in the presence of (i) is equivalent to (ii′): S is separable as an R-algebra, finitely generated and projective as an R-module, and the fixed ring under the group of all R-algebra automorphisms of S is exactly R.

Remarks on Galois cohomology and definability

1997 ◽  
Vol 62 (2) ◽  
pp. 487-492 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Automorphism Group ◽  
Model Theory ◽  
Galois Theory ◽  
Homogeneous Spaces ◽  
Galois Cohomology ◽  
Seminal Paper ◽  
Descent Theory ◽  
Definable Sets ◽  
Basic Features ◽  
Special Case

In this paper we develop some basic features of Galois cohomology, specifically the connection between first Galois cohomology groups and principal homogeneous spaces, in a model-theoretic context. “Descent theory” also fits into our approach.The model theory involved is elementary, and the reader is referred to [2]. It should be said that we make crucial use of Meq in our analysis. The reader is also referred to Poizat's seminal paper “Une theorie de Galois imaginaire” ([6]). Although our results do not depend on Poizat's work, it is in his paper that the model-theoretic context is suggested for a generalised treatment of Galois theory.Nothing in this paper is particularly deep. We are concerned mainly with translating between the Galois cohomological language and the language of definable sets and definable families of definable sets. We will introduce (in a suitable context) the notion of a definable cocycle (from an automorphism group to a definable group G). The (classical) situation of profinite and continuous cocycles will be a special case. Kolchin's theory of constrained cohomology will be another special case, and our results yield a substantially simpler proof of his Theorem 5 from Chapter VII of [4]. In any case model-theorists will see that definable cocycles correspond to objects with which they are already quite familiar—commuting families of definable bijections.

Field Extensions and Galois Theory

1984 ◽  
Author(s):  
Julio R. Bastida ◽  
Roger Lyndon
Keyword(s):  
Galois Theory ◽  
Field Extensions
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