The center of the endomorphism ring of a split mixed abelian group

1999 ◽  
Vol 40 (5) ◽  
pp. 907-916 ◽  
Author(s):  
P. A. Krylov ◽  
E. D. Klassen
Keyword(s):  
Abelian Group ◽  
Endomorphism Ring ◽  
Mixed Abelian Group

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.

QUASI-ISOMORPHISMS AND GROUPS OF QUASI-HOMOMORPHISMS

2009 ◽  
Vol 08 (05) ◽  
pp. 617-627
Author(s):  
ULRICH ALBRECHT ◽  
SIMION BREAZ
Keyword(s):  
Abelian Group ◽  
Dual Problem ◽  
Abelian Groups ◽  
Mixed Abelian Group ◽  
Free Rank ◽  
Mixed Groups ◽  
Torsion Free Rank ◽  
Torsion Free

This paper investigates to which extent a self-small mixed Abelian group G of finite torsion-free rank is determined by the groups Hom (G,C) where C is chosen from a suitable class [Formula: see text] of Abelian groups. We show that G is determined up to quasi-isomorphism if [Formula: see text] is the class of all self-small mixed groups C with r0(C) ≤ r0(G). Several related results are given, and the dual problem of orthogonal classes is investigated.

scholarly journals On the lattice of right ideals of the endomorphism ring of an abelian group

1988 ◽  
Vol 38 (2) ◽  
pp. 273-291 ◽  
Author(s):  
Theodore G. Faticoni
Keyword(s):  
Abelian Group ◽  
Finite Index ◽  
Endomorphism Ring ◽  
Finite Rank ◽  
Ideal Structure ◽  
Left Module ◽  
Linear Topology ◽  
Closed Submodules ◽  
The Right ◽  
Torsion Free

Let A be an abelian group, let ∧ = End (A), and assume that A is a flat left ∧-module. Then σ = { right ideals I ⊂ ∧ | IA = A} generates a linear topology oil ∧. We prove that Hom(A,·) is an equivalence from the category of those groups B ⊂ An satisfying B = Hom(A, B)A, onto the category of σ-closed submodules of finitely generated free right ∧-modules. Applications classify the right ideal structure of A, and classify torsion-free groups A of finite rank which are (nearly) isomorphic to each A-generated subgroup of finite index in A.

scholarly journals The Baer–Kaplansky Theorem for All Abelian Groups and Modules

2021 ◽  
pp. 1-12
Author(s):  
Simion Breaz ◽  
Tomasz Brzeziński
Keyword(s):  
Abelian Group ◽  
Endomorphism Ring ◽  
Abelian Groups

It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups [Formula: see text] and [Formula: see text] is induced by an isomorphism between [Formula: see text] and [Formula: see text] and an element from [Formula: see text]. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module [Formula: see text] determines [Formula: see text] as a module over its endomorphism ring.

Sign in / Sign up

Export Citation Format

Share Document