Groups Admitting only Finitely Many Nilpotent Ring Structures

1986 ◽  
Vol 29 (2) ◽  
pp. 197-203
Author(s):  
Shalom Feigelstock
Keyword(s):  
Abelian Groups ◽  
Complete Classification ◽  
Actual Number ◽  
Nilpotent Ring ◽  
Ring Structures ◽  
Free Case ◽  
Torsion Free

AbstractThe abelian groups which are the additive groups of only finitely many non-isomorphic (associative) nilpotent rings are studied. Progress is made toward a complete classification of these groups. In the torsion free case, the actual number of non-isomorphic nilpotent rings these groups support is obtained.

scholarly journals CONVEXLY ORDERABLE GROUPS AND VALUED FIELDS

2014 ◽  
Vol 79 (01) ◽  
pp. 154-170
Author(s):  
JOSEPH FLENNER ◽  
VINCENT GUINGONA
Keyword(s):  
Abelian Groups ◽  
Complete Classification ◽  
Necessary Condition ◽  
Valued Fields ◽  
Ordered Groups ◽  
Algebraic Theories ◽  
Convexly Orderable

Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.

scholarly journals Sub-prime radical classes determined by zerorings

1975 ◽  
Vol 12 (1) ◽  
pp. 95-97 ◽  
Author(s):  
B.J. Gardner
Keyword(s):  
Abelian Groups ◽  
Complete Classification ◽  
Prime Radical ◽  
Radical Class

It is shown that the correspondence which associates with each radical class τ of abelian groups the (radical) class of prime radical rings with additive groups in τ gives a complete classification of those radical classes of rings which are determined (as lower radicals) by zerorings.

Model-Completeness and Elementary Properties of Torsion Free Abelian Groups

1974 ◽  
Vol 26 (4) ◽  
pp. 829-840
Author(s):  
Elias Zakon
Keyword(s):  
High Power ◽  
Abelian Groups ◽  
Elementary Theory ◽  
Complete Classification ◽  
Formal Logic ◽  
Predicate Calculus ◽  
Symbolic Logic ◽  
Model Completeness ◽  
Free Abelian Groups ◽  
Torsion Free

The decidability of the elementary theory of abelian groups, and their complete classification by elementary properties (i.e. those formalizable in the lower predicate calculus (LPC) of formal logic), were established by W. Szmielew [13]. More general results were proved by Eklof and Fischer [2], and G. Sabbagh [12]. The rather formidable "high-power" techniques used in obtaining these remarkable results, and the length of the proofs (W. Szmielew's proof takes about 70 pages) triggered off several attempts at simplification. M. I. Kargapolov's proof [3] unfortunately turned out to be erroneous (cf. J. Mennicke's review in the Journal of Symbolic Logic, vol. 32, p. 535).

Orderings in locally compact Abelian groups and the theorem of F. and M. Riesz

1983 ◽  
Vol 93 (3) ◽  
pp. 441-457 ◽  
Author(s):  
Edwin Hewitt ◽  
Shozo Koshi
Keyword(s):  
Harmonic Analysis ◽  
Abelian Groups ◽  
Complete Classification ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Background (1·1). Ordered Abelian groups have been studied for nearly a century. Since the early 1950's, it has been recognized that orderings in locally compact Abelian groups can play an important rôle in harmonic analysis on such groups. In this paper we study orderings, especially in topological Abelian groups with either topological or measure-theoretic properties, obtaining nearly a complete classification of such orderings. We then apply these results to determine the limitations of the celebrated theorem of F. and M. Riesz on such groups.

A Problem on Relative Projectivity for Abelian Groups

1986 ◽  
Vol 29 (1) ◽  
pp. 114-122 ◽  
Author(s):  
S. Feigelstock ◽  
R. Raphael
Keyword(s):  
Abelian Groups ◽  
Free Groups ◽  
Mixed Case ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Free Case ◽  
Torsion Groups ◽  
Torsion Free ◽  
Whitehead Groups

AbstractThe article studies the class of abelian groups G such that in every direct sum decomposition G = A ⊕ B, A is 5-projective. Such groups are called pds groups and they properly include the quasi-projective groups.The pds torsion groups are fully determined.The torsion-free case depends on a lemma that establishes freedom in the non-indecomposable case for several classes of groups. There is evidence suggesting freedom in the general reduced torsion-free case but this is not established and prompts a logical discussion. It is shown, for example, that pds torsion-free groups must be Whitehead if they are not indecomposable, but that there exists Whitehead groups that are not pds if there exist non-free Whitehead groups.The mixed case is characterized and examples are given.

scholarly journals Products of locally cyclic groups

2021 ◽  
Author(s):  
Bernhard Amberg ◽  
Yaroslav Sysak
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Complete Classification ◽  
The Other ◽  
Supersoluble Group ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Prüfer Rank ◽  
Torsion Free

AbstractWe consider groups of the form $${G} = {AB}$$ G = AB with two locally cyclic subgroups A and B. The structure of these groups is determined in the cases when A and B are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case where A and B are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic subgroups. As an application, it is shown that the Prüfer rank of a periodic product of two locally cyclic subgroups does not exceed 3, and this bound is sharp. It is also proved that a product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group. This generalizes a well-known theorem of B. Huppert for finite groups.

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