von Neumann regular f-rings

1999 ◽  
pp. 201-208
Author(s):  
Niels Schwartz ◽  
James J. Madden
Keyword(s):  
Von Neumann ◽  
Von Neumann Regular

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

On (Strongly) Gorenstein Von Neumann Regular Rings

2011 ◽  
Vol 39 (9) ◽  
pp. 3242-3252 ◽  
Author(s):  
Najib Mahdou ◽  
Mohammed Tamekkante ◽  
Siamak Yassemi
Keyword(s):  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

1988 ◽  
Vol 53 (4) ◽  
pp. 1177-1187
Author(s):  
W. A. MacCaull
Keyword(s):  
Intuitionistic Logic ◽  
Main Idea ◽  
Rational Functions ◽  
Completeness Theorem ◽  
Point Of View ◽  
Polynomial Rings ◽  
Von Neumann ◽  
Ordered Fields ◽  
Von Neumann Regular ◽  
Theoretic Point

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

scholarly journals RINGS OF DEFINABLE SCALARS OF VERMA MODULES

2007 ◽  
Vol 06 (05) ◽  
pp. 779-787 ◽  
Author(s):  
SONIA L'INNOCENTE ◽  
MIKE PREST
Keyword(s):  
Lie Algebra ◽  
Model Theory ◽  
Regular Ring ◽  
Verma Module ◽  
Closed Field ◽  
Verma Modules ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Canonical Map ◽  
Von Neumann Regular

Let M be a Verma module over the Lie algebra, sl 2(k), of trace zero 2 × 2 matrices over the algebraically closed field k. We show that the ring, RM, of definable scalars of M is a von Neumann regular ring and that the canonical map from U( sl 2(k)) to RM is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules.

scholarly journals Normal Subgroups of Classical Groups over von Neumann Regular Rings

1994 ◽  
Vol 169 (3) ◽  
pp. 863-873
Author(s):  
F.A. Arlinghaus ◽  
L.N. Vaserstein ◽  
H. You
Keyword(s):  
Classical Groups ◽  
Normal Subgroups ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

Strongly 0-dimensional Modules

2014 ◽  
Vol 57 (1) ◽  
pp. 159-165 ◽  
Author(s):  
Kürşat Hakan Oral ◽  
Neslihan Ayşen Özkirişci ◽  
Ünsal Tekir
Keyword(s):  
Multiplication Module ◽  
Finite Intersection ◽  
Von Neumann ◽  
Prime Submodules ◽  
Von Neumann Regular ◽  
Prime Submodule ◽  
Dimensional Module

AbstractIn a multiplication module, prime submodules have the following property: if a prime submodule contains a finite intersection of submodules, then one of the submodules is contained in the prime submodule. In this paper, we generalize this property to infinite intersection of submodules and call such prime submodules strongly prime submodules. A multiplication module in which every prime submodule is strongly prime will be called a strongly 0-dimensional module. It is also an extension of strongly 0-dimensional rings. After this we investigate properties of strongly 0-dimensional modules and give relations of von Neumann regular modules, Q-modules and strongly 0-dimensional modules

Partial orders on the power sets of Baer rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050011 ◽  
Author(s):  
B. Ungor ◽  
S. Halicioglu ◽  
A. Harmanci ◽  
J. Marovt
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Baer Ring ◽  
Power Set ◽  
Von Neumann Regular ◽  
Baer Rings ◽  
The Right

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

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