scholarly journals Locally nilpotent skew linear groups

1986 ◽  
Vol 29 (1) ◽  
pp. 101-113 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Positive Integer ◽  
Division Ring ◽  
Matrix Ring ◽  
Linear Groups ◽  
Full Matrix ◽  
Locally Nilpotent ◽  
Completely Reducible ◽  
Full Matrix Ring

Throughout this paper D denotes a division ring with centre F and n a positive integer. A subgroup G of GL(n,D) is absolutely irreducible if the F-subalgebra F[G] enerated by G is the full matrix ring Dn ×n. It is completely reducible (resp. irreducible) if row n-space Dn over D is completely reducible (resp. irreducible), as D–G bimodule in the obvious way. Absolutely irreducible skew linear groups have a more restricted structure than irreducible skew linear groups, see for example [7],[8], [8] and [10]. Here we make a start on elucidating the structure of locally nilpotent suchgroups.

Soluble-by-periodic skew linear groups

1984 ◽  
Vol 96 (3) ◽  
pp. 379-389 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Division Ring ◽  
Matrix Algebra ◽  
Locally Finite Group ◽  
Linear Groups ◽  
Locally Finite ◽  
Full Matrix ◽  
Full Matrix Algebra ◽  
Finite Image

Let D be a division ring with central subfield F, n a positive integer and G a subgroup of GL(n, D) such that the F-subalgebra F[G] generated by G is the full matrix algebra Dn×n. If G is soluble then Snider [9] proves that G is abelian by locally finite. He also shows that this locally finite image of G can be any locally finite group. Of course not every abelian by locally finite group is soluble. This suggests that Snider's conclusion should apply to some wider class of groups.

Representation of tiled matrix rings as full matrix rings

1989 ◽  
Vol 105 (1) ◽  
pp. 67-72 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Positive Integer ◽  
Jacobson Radical ◽  
Integral Domain ◽  
Matrix Ring ◽  
The Other ◽  
Matrix Rings ◽  
Full Matrix ◽  
Image Position ◽  
Full Matrix Ring ◽  
Unique Maximal Ideal

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and setThen T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).

scholarly journals Strong Regularity in Arbitrary Rings

1952 ◽  
Vol 4 ◽  
pp. 51-53 ◽  
Author(s):  
Tetsuo Kandô
Keyword(s):  
Matrix Ring ◽  
Strong Regularity ◽  
Full Matrix ◽  
Regular Ideal ◽  
Strongly Regular ◽  
Full Matrix Ring

An element a of a ring R is called regular, if there exists an element x of R such that a×a = a, and a two-sided ideal a in R is said to be regular if each of its elements is regular B. Brown and N. H. McCoy [1] has recently proved that every ring R has a unique maximal regular two-sided ideal M(R), and that M(R) has the following radical-like property: (i) M(R/M(R)) = 0; (ii) if a is a two-sided ideal of R, then M(a) = a ∩ M(R); (iii) M(Rn) = (M(R))n, where Rn denotes a full matrix ring of order n over R. Arens and Kaplansky [2] has defined an element a of R to be strongly regular when there exists an element x of R such that a2x = a. We shall prove in this note that replacing “regularity” by “strong regularity,” we have also a unique maximal strongly regular ideal N(R), and shall investigate some of its properties.

Automorphisms of the zero-divisor graph of the full matrix ring

2016 ◽  
Vol 65 (5) ◽  
pp. 991-1002 ◽  
Author(s):  
Jinming Zhou ◽  
Dein Wong ◽  
Xiaobin Ma
Keyword(s):  
Zero Divisor ◽  
Matrix Ring ◽  
Full Matrix ◽  
Zero Divisor Graph ◽  
Full Matrix Ring

scholarly journals Irreducible locally nilpotent finitary skew linear groups

1995 ◽  
Vol 38 (1) ◽  
pp. 63-76 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Linear Group ◽  
Division Ring ◽  
Linear Groups ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finitary Linear Group ◽  
Locally Nilpotent ◽  
Finitary Linear ◽  
Finite Dimensional Case

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dimDV(g-1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dimDV is infinite then dimDV = |G| = ℵ0 and G is a locally finite q-group for some prime q ≠ char D. Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.

Density of Polynomial Maps

2010 ◽  
Vol 53 (2) ◽  
pp. 223-229 ◽  
Author(s):  
Chen-Lian Chuang ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Prime Ring ◽  
Division Ring ◽  
Polynomial Identity ◽  
Matrix Ring ◽  
Polynomial Maps ◽  
Finite Matrix ◽  
Nonzero Polynomial

AbstractLet R be a dense subring of End(DV), where V is a left vector space over a division ring D. If dimDV = ∞, then the range of any nonzero polynomial ƒ (X1, … , Xm) on R is dense in End(DV). As an application, let R be a prime ring without nonzero nil one-sided ideals and 0 ≠ a ∈ R. If a f (x1, … , xm)n(xi) = 0 for all x1, … , xm ∈ R, where n(xi ) is a positive integer depending on x1, … , xm, then ƒ (X1, … , Xm) is a polynomial identity of R unless R is a finite matrix ring over a finite field.

π-Baer rings

2018 ◽  
Vol 17 (02) ◽  
pp. 1850029 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Yeliz Kara ◽  
Adnan Tercan
Keyword(s):  
Left Ideal ◽  
Matrix Ring ◽  
Idempotent Element ◽  
Matrix Rings ◽  
Full Matrix ◽  
Baer Ring ◽  
Baer Rings ◽  
Base Ring ◽  
Full Matrix Ring ◽  
The Right

We say a ring [Formula: see text] is [Formula: see text]-Baer if the right annihilator of every projection invariant left ideal of [Formula: see text] is generated by an idempotent element of [Formula: see text]. In this paper, we study connections between the [Formula: see text]-Baer condition and related conditions such as the Baer, quasi-Baer and [Formula: see text]-extending conditions. The [Formula: see text]-by-[Formula: see text] generalized triangular and the [Formula: see text]-by-[Formula: see text] triangular [Formula: see text]-Baer matrix rings are characterized. Also, we prove that a [Formula: see text]-by-[Formula: see text] full matrix ring over a [Formula: see text]-Baer ring is a [Formula: see text]-Baer ring. In contrast to the Baer condition, it is shown that the [Formula: see text]-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.

scholarly journals The moduli of representations with Borel mold

2014 ◽  
Vol 25 (07) ◽  
pp. 1450067
Author(s):  
Kazunori Nakamoto
Keyword(s):  
Matrix Ring ◽  
Triangular Matrices ◽  
Full Matrix ◽  
Upper Triangular Matrices ◽  
Inner Automorphisms ◽  
Full Matrix Ring

The author constructs the moduli of representations whose images generate the subalgebra of upper triangular matrices (up to inner automorphisms) of the full matrix ring for any groups and any monoids.

Products of idempotent integer matrices

1991 ◽  
Vol 110 (3) ◽  
pp. 431-441 ◽  
Author(s):  
John Fountain
Keyword(s):  
Valuation Ring ◽  
Matrix Ring ◽  
Discrete Valuation Ring ◽  
Sufficient Condition ◽  
Principal Ideal Domain ◽  
Necessary And Sufficient Condition ◽  
Full Matrix ◽  
Ring Of Integers ◽  
Full Matrix Ring ◽  
Necessary And Sufficient

AbstractLet E denote the set of non-identity idempotent matrices in the full matrix ring Mn(R) over a principal ideal domain R. A necessary and sufficient condition is found for the subsemigroup generated by E to be the set of all matrices in Mn(R) of rank less than n. The condition is satisfied when R is a discrete valuation ring and when R is the ring of integers. Thus every n × n matrix of rank less than n is a product of idempotent integer matrices.

Sign in / Sign up

Export Citation Format

Share Document