scholarly journals The endomorphism ring of a locally free module

1983 ◽  
Vol 35 (3) ◽  
pp. 308-326 ◽  
Author(s):  
W. N. Franzsen ◽  
P. Schultz
Keyword(s):  
Large Class ◽  
Endomorphism Ring ◽  
Special Class ◽  
Free Module ◽  
Endomorphism Rings ◽  
Free Modules

AbstractWe identify a large class of rings over which locally free modules are determined by their endomorphism rings. We characterize these endomorphism rings and consider under what circumstances the conditions on the locally free modules can be relaxed, for example by requiring that only one of the rings need be in the special class, or by replacing ‘free' by “projective”.

scholarly journals The characterization problem for endomorphism rings

1991 ◽  
Vol 50 (1) ◽  
pp. 116-137 ◽  
Author(s):  
J. L. García
Keyword(s):  
Endomorphism Ring ◽  
Free Module ◽  
Endomorphism Rings ◽  
Characterization Problem ◽  
Algebraic Properties

AbstractWe consider the problem of characterizing by abstract properties the rings which are isomorphic to the endomorphism ring End (RF) of some free module F over a ring R in a given class R of rings. We solve this problem when R is any class of rings (by employing topological notions) and when R is the class of all the left Kasch rings (in terms of algebraic properties only).

Baer Endomorphism Rings and Closure Operators

1978 ◽  
Vol 30 (5) ◽  
pp. 1070-1078 ◽  
Author(s):  
Soumaya M. Khuri
Keyword(s):  
Endomorphism Ring ◽  
Free Module ◽  
Linear Transformations ◽  
Endomorphism Rings ◽  
Dual Vector ◽  
Annihilator Ideal ◽  
Baer Ring ◽  
Left Annihilator ◽  
Torsion Free ◽  
Made In

A Baer ring is a ring in which every right (and left) annihilator ideal is generated by an idempotent. Generalizing quite naturally from the fact that the endomorphism ring of a vector space is a Baer ring, Wolfson [5; 6] investigated questions such as when the endomorphism ring of a free module is a Baer ring, and when the ring of continuous linear transformations on a pair of dual vector spaces is a Baer ring. A further generalization was made in [7], where the question of when the endomorphism ring of a torsion-free module over a semiprime left Goldie ring is a Baer ring was treated.

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.

Isomorphisms of Prime Goldie Semi-Principal Left Ideal Rings, II

1988 ◽  
Vol 31 (3) ◽  
pp. 374-379 ◽  
Author(s):  
Kenneth G. Wolfson
Keyword(s):  
Left Ideal ◽  
Endomorphism Ring ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
Free Module ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Ore Domain ◽  
Necessary And Sufficient

AbstractA prime Goldie ring K, in which each finitely generated left ideal is principal is the endomorphism ring E(F, A) of a free module A, of finite rank, over an Ore domain F. We determine necessary and sufficient conditions to insure that whenever K ≅ E(F, A) ≅ E(G, B) (with A and B free and finitely generated over domains F and G) then (F, A) is semi-linearly isomorphic to (G, B). We also show, by example, that it is possible for K ≅ E(F, A ) ≅ E(G, B), with F and G, not isomorphic.

scholarly journals Isomorphisms between endomorphism rings of projective modules

1993 ◽  
Vol 35 (3) ◽  
pp. 353-355 ◽  
Author(s):  
José Luis García ◽  
Juan Jacobo Simón
Keyword(s):  
Morita Equivalence ◽  
Projective Modules ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Morita Equivalent ◽  
Free Modules

Let R and S be arbitrary rings, RM and SN countably generated free modules, and let φ:End(RM)→End(sN) be an isomorphism between the endomorphism rings of M and N. Camillo [3] showed in 1984 that these assumptions imply that R and S are Morita equivalent rings. Indeed, as Bolla pointed out in [2], in this case the isomorphism φ must be induced by some Morita equivalence between R and S. The same holds true if one assumes that RM and SN are, more generally, non-finitely generated free modules.

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.

(m,n)-Injective and (m,n)-coherent endomorphism rings of modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050048
Author(s):  
Lixin Mao
Keyword(s):  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Text Coherence ◽  
Coherent Ring ◽  
The Right ◽  
Positive Integers ◽  
Finitely Presented

Let [Formula: see text] and [Formula: see text] be fixed positive integers. [Formula: see text] is called a right [Formula: see text]-injective ring if every right [Formula: see text]-homomorphism from an [Formula: see text]-generated submodule of the right [Formula: see text]-module [Formula: see text] to [Formula: see text] extends to one from [Formula: see text] to [Formula: see text]; [Formula: see text] is called a right [Formula: see text]-coherent ring if each [Formula: see text]-generated submodule of the right [Formula: see text]-module [Formula: see text] is a finitely presented right [Formula: see text]-module. Let [Formula: see text] be a right [Formula: see text]-module. We study the [Formula: see text]-injectivity and [Formula: see text]-coherence of the endomorphism ring [Formula: see text] of [Formula: see text]. Some applications are also given.

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