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.

scholarly journals Free division rings of fractions of crossed products of groups with Conradian left-orders

2020 ◽  
Vol 32 (3) ◽  
pp. 739-772
Author(s):  
Joachim Gräter
Keyword(s):  
Vector Space ◽  
Endomorphism Ring ◽  
Division Ring ◽  
Skew Field ◽  
Division Rings ◽  
Products Of Groups ◽  
Ring Of Fractions ◽  
Left Order ◽  
The Right ◽  
Formal Power

AbstractLet D be a division ring of fractions of a crossed product {F[G,\eta,\alpha]}, where F is a skew field and G is a group with Conradian left-order {\leq}. For D we introduce the notion of freeness with respect to {\leq} and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space {F((G))} of all formal power series in G over F with respect to {\leq}. From this we obtain that all division rings of fractions of {F[G,\eta,\alpha]} which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, {F[G,\eta,\alpha]} possesses a division ring of fraction which is free in this sense if and only if the rational closure of {F[G,\eta,\alpha]} in the endomorphism ring of the corresponding right F-vector space {F((G))} is a skew field.

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).

scholarly journals The endomorphism ring of a finite-length module

1989 ◽  
Vol 39 (1) ◽  
pp. 141-143
Author(s):  
Rainer Schulz
Keyword(s):  
Finite Length ◽  
Endomorphism Ring ◽  
Isomorphism Type ◽  
Loewy Length

Let M be an R-module of finite length. For a simple R-module A, let ℓA denote the nuber of times the isomorphism type of A appears in a composition chain of M, and let σ denote the maxinium of the ℓA, A ranging over all simple submodules of M. Let S be the endomorphism ring of M. We show that the Loewy length of S is bounded by σ.

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.

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.

scholarly journals Quotient rings of semiprime rings with bounded index

1982 ◽  
Vol 23 (1) ◽  
pp. 53-64 ◽  
Author(s):  
John Hannah
Keyword(s):  
Direct Product ◽  
Polynomial Identity ◽  
Semiprime Ring ◽  
Quotient Ring ◽  
Matrix Rings ◽  
Special Cases ◽  
Semiprime Rings ◽  
Strongly Regular ◽  
Quotient Rings ◽  
Finite Direct Product

We say that a ring R has bounded index if there is a positive integer n such that an = 0 for each nilpotent element a of R. If n is the least such integer we say R has index n. For example, any semiprime right Goldie ring has bounded index, and so does any semiprime ring satisfying a polynomial identity [10, Theorem 10.8.2]. This paper is mainly concerned with the maximal (right) quotient ring Q of a semiprime ring R with bounded index. Several special cases of this situation have already received attention in the literature. If R satisfies a polynomial identity [1], or if every nonzero right ideal of R contains a nonzero idempotent [18] then it is known that Q is a finite direct product of matrix rings over strongly regular self-injective rings, the size of the matrices being bounded by the index of R. On the other hand if R is reduced (that is, has index 1) then Q is a direct product of a strongly regular self-injective ring and a biregular right self-injective ring of type III ([2] and [15]; the terminology is explained in [6]). We prove the following generalization of these results (see Theorems 9 and 11).

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.

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