scholarly journals Endomorphism rings of p-local finite spectra are semi-perfect

2009 ◽  
Vol 139 (3) ◽  
pp. 567-574 ◽  
Author(s):  
Petar Pavešić
Keyword(s):  
Endomorphism Ring ◽  
Jacobson Radical ◽  
Strong Form ◽  
Main Step ◽  
Unique Factorization ◽  
Endomorphism Rings ◽  
Finite Spectrum ◽  
Geometric Properties ◽  
Perfect Ring ◽  
Algebraic Properties

Let X be a finite spectrum. We prove that R(X(p)), the endomorphism ring of the p-localization of X, is a semi-perfect ring. This implies, among other things, a strong form of unique factorization for finite p-local spectra. The main step in the proof is that the Jacobson radical of R(X(p)) is idempotent-lifting, which is proved by a combination of geometric properties of finite spectra and algebraic properties of the p-localization.

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).

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.

Modules Whose Endomorphism Rings Are (m, n)-Coherent

2019 ◽  
Vol 26 (02) ◽  
pp. 231-242
Author(s):  
Xiaoqiang Luo ◽  
Lixin Mao
Keyword(s):  
Left Ideal ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Coherent Ring ◽  
Left Annihilator

Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator [Formula: see text] is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of Mn if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.

Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields

2007 ◽  
Vol 50 (3) ◽  
pp. 409-417 ◽  
Author(s):  
Florian Luca ◽  
Igor E. Shparlinski
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Complex Multiplication ◽  
Number Fields ◽  
Quadratic Number ◽  
Endomorphism Rings ◽  
Quadratic Number Field ◽  
Curves Over Finite Fields

AbstractWe show that, for most of the elliptic curves E over a prime finite field p of p elements, the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over p is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over p.

Endomorphism Rings of Quasi-Injective Modules

1968 ◽  
Vol 20 ◽  
pp. 895-903 ◽  
Author(s):  
B. L. Osofsky
Keyword(s):  
Jacobson Radical ◽  
Essential Extension ◽  
Endomorphism Rings ◽  
Injective Modules

Y. Utumi (14 and 15) obtained some interesting results on self-injective rings. He showed that, if R is right self-injective, then so is R/J, where J is the Jacobson radical of R. Also, if R is right self-injective and regular, then R is left self-injective for any set of orthogonal idempotents is an essential extension of . This note extends these results to endomorphism rings of quasi-injective modules.

On PP-Endomorphism Rings

1993 ◽  
Vol 36 (2) ◽  
pp. 227-230 ◽  
Author(s):  
W. K. Nicholson
Keyword(s):  
Endomorphism Ring ◽  
Necessary Condition ◽  
Endomorphism Rings ◽  
Left Module ◽  
Pp Ring

AbstractA characterization is given of when all kernels (respectively images) of endomorphisms of a module are direct summands, a necessary condition being that the endomorphism ring itself is a left (respectively right) PP-ring. This result generalizes theorems of Small, Lenzing and Colby-Rutter and shows that R is left hereditary if and only if the endomorphism ring of every injective left module is a right PP-ring.

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.

On rings whose nonunits are a unit multiple of a nilpotent

2016 ◽  
Vol 16 (07) ◽  
pp. 1750140
Author(s):  
Peter Vámos
Keyword(s):  
Vector Space ◽  
Finite Length ◽  
Endomorphism Ring ◽  
Polynomial Identity ◽  
Morita Equivalence ◽  
Complete Characterization ◽  
Krull Dimension ◽  
Endomorphism Rings ◽  
Matrix Rings ◽  
Skew Field

The rings in the title were called UN rings by Călugăreanu in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182]. He gave two examples of simple UN rings: matrix rings over a skew field and a ring, which is the filtered union of such rings. We give new examples of simple UN rings as endomorphism rings of ‘vector space like’ modules and determine the structure of UN rings, which satisfy a polynomial identity or have Krull dimension. We also answer some questions in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182] about Morita equivalence of UN rings and show that this question is related to Köthe’s conjecture. Finally a complete characterization is given of modules over a Dedekind domain (in particular Abelian groups) and modules of finite length with a UN endomorphism ring.

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