scholarly journals Bounded topologies on endomorphism rings

2011 ◽  
Vol 20 (1) ◽  
pp. 1-3
Author(s):  
HOREA F. ABRUDAN ◽  
Keyword(s):  
Abelian Group ◽  
Endomorphism Rings ◽  
Ring Topology ◽  
Left And Right ◽  
Ring Of Endomorphisms

We prove in this note that the ring of endomorphisms of an infinite bounded Abelian group admits a nondiscrete right bounded ring topology. We give an example of an Abelian group whose ring of endomorphisms admits both nondiscrete left and right bounded topologies but does not admit a nondiscrete bounded ring topology.

Permutation endomorphisms and refinement of a theorem of Birkhoff

1960 ◽  
Vol 56 (4) ◽  
pp. 322-328 ◽  
Author(s):  
H. K. Farahat ◽  
L. Mirsky
Keyword(s):  
Abelian Group ◽  
Finite Number ◽  
Linear Combination ◽  
Finite Linear Combination ◽  
Usual Manner ◽  
Restricted Permutation ◽  
Image Position ◽  
Ring Of Endomorphisms ◽  
Finite Linear

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equationsA permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

Commutative Endomorphism Rings

1971 ◽  
Vol 23 (1) ◽  
pp. 69-76 ◽  
Author(s):  
J. Zelmanowitz
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Prime Rings ◽  
Infinite Cyclic Group ◽  
Free Abelian Groups ◽  
Scalar Action ◽  
Torsion Free

The problem of classifying the torsion-free abelian groups with commutative endomorphism rings appears as Fuchs’ problems in [4, Problems 46 and 47]. They are far from solved, and the obstacles to a solution appear formidable (see [4; 5]). It is, however, easy to see that the only dualizable abelian group with a commutative endomorphism ring is the infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class of prime rings R which possess a dualizable module M with a commutative endomorphism ring. A characterization of such rings is obtained in § 6, which as would be expected, places stringent restrictions on the ring and the module.Throughout we will write homomorphisms of modules on the side opposite to the scalar action. Rings will not be assumed to contain identity elements unless otherwise indicated.

scholarly journals Completely simple endomorphism rings of modules

2018 ◽  
Vol 19 (2) ◽  
pp. 223
Author(s):  
Victor Bovdi ◽  
Mohamed Salim ◽  
Mihail Ursul
Keyword(s):  
Commutative Rings ◽  
Compact Ring ◽  
Endomorphism Rings ◽  
Locally Compact ◽  
Ring Topology ◽  
Simple Ring ◽  
Finite Topology ◽  
Completely Simple ◽  
The Ideal

<p>It is proved that if A<sub>p</sub> is a countable elementary abelian p-group, then: (i) The ring End (A<sub>p</sub>) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End (A<sub>p</sub>)/I, where I is the ideal of End (A<sub>p</sub>) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End (A<sub>p</sub>) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained.</p>

Linear diquasigroups

2017 ◽  
Vol 16 (03) ◽  
pp. 1750057
Author(s):  
Bokhee Im ◽  
K. Matczak ◽  
J. D. H. Smith
Keyword(s):  
Abelian Group ◽  
Differential Calculus ◽  
Sufficient Condition ◽  
Left And Right ◽  
Algebraic Language

Following the prototype of dimonoids, diquasigroups are directed versions of quasigroups, where the structure is split into left and right quasigroups on the same set. The linear and affine diquasigroups that form the topic of this paper are built on the foundation of a module. In this context, various issues that may be difficult to handle in the general case, for example identification of the largest two-sided quasigroup image, become more tractable. An appropriate universal algebraic language for affine diquasigroups is established, and the entropic models of this language are characterized. Various interesting classes of linear and affine diquasigroups are singled out for special attention, such as internally associative, Bol, and symmetric diquasigroups. The problem of determining which linear diquasigroups have an abelian group as their undirected replica is raised. One sufficient condition is provided, formulated in terms of a differential calculus for one-sided quasigroup words.

scholarly journals An extension of the Krull-Schmidt theorem

1969 ◽  
Vol 1 (1) ◽  
pp. 109-114 ◽  
Author(s):  
S. B. Conlon
Keyword(s):  
Local Ring ◽  
Partial Answer ◽  
Endomorphism Rings ◽  
Abelian Categories ◽  
Ring Of Endomorphisms

In the usual Krull-Schmidt-Azumaya theorem in abelian categories it is essential that each of the direct summands has a local ring of endomorphisms. A partial answer is given here to the case where this last condition is not satisfied by the indecomposable direct summands. It is found that those summands with local endomorphism rings are determined up to isomorphism and cardinality of occurrence.

scholarly journals Modules with the quasi-summand intersection property

1991 ◽  
Vol 44 (2) ◽  
pp. 189-201 ◽  
Author(s):  
Ulrich Albrecht ◽  
Jutta Hausen
Keyword(s):  
Abelian Group ◽  
Endomorphism Ring ◽  
Free Abelian Group ◽  
Intersection Property ◽  
Endomorphism Rings ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Free Modules ◽  
Torsion Free ◽  
Complete Characterisation

Given a torsion-free abelian group G, a subgroup A of G is said to be a quasi-summand of G if nG ≤ A ⊕ B ≤ G for some subgroup B of G and some positive integer n. If the intersection of any two quasi-summands of G is a quasi-summand, then G is said to have the quasi-summand intersection property. This is a generalisation of the summand intersection property of L. Fuchs. In this note, we give a complete characterisation of the torsion-free abelian groups (in fact, torsion-free modules over torsion-free rings) with the quasi-summand intersection property. It is shown that such a characterisation cannot be given via endomorphism rings alone but must involve the way in which the endomorphism ring acts on the underlying group. For torsion-free groups G of finite rank without proper fully invariant quasi-summands however, the structure of its quasi-endomorphism ring QE(G) suffices: G has the quasi-summand intersection property if and only if the ring QE(G) is simple or else G is strongly indecomposable.

scholarly journals Finite torsion-free rank endomorphism rings

2015 ◽  
Vol 31 (1) ◽  
pp. 39-43
Author(s):  
SIMION BREAZ ◽  
Keyword(s):  
Abelian Group ◽  
Small Group ◽  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Direct Sum ◽  
Bounded Group ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

We prove that a finite torsion-free rank abelian group with finite torsion-free rank endomorphism ring is a direct sum of a bounded group and a self-small group.

scholarly journals Abelian groups that are torsion over their endomorphism rings

2001 ◽  
Vol 64 (2) ◽  
pp. 255-263
Author(s):  
J. Hill ◽  
P. Hill ◽  
W. Ullery
Keyword(s):  
Abelian Group ◽  
Free Group ◽  
Abelian Groups ◽  
Torsion Theory ◽  
Endomorphism Rings ◽  
Torsion Free Group ◽  
Large Classes ◽  
Infinite Rank ◽  
Torsion Modules ◽  
Torsion Free

Using Lambek torsion as the torsion theory, we investigate the question of when an Abelian group G is torsion as a module over its endomorphism ring E. Groups that are torsion modules in this sense are called ℒ-torsion. Among the classes of torsion and truly mixed Abelian groups, we are able to determine completely those groups that are ℒ-torsion. However, the case when G is torsion free is more complicated. Whereas no torsion-free group of finite rank is ℒ-torsion, we show that there are large classes of torsion-free groups of infinite rank that are ℒ-torsion. Nevertheless, meaningful definitive criteria for a torsion-free group to be ℒ-torsion have not been found.

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