Homomorphisms of Matrix Semigroups over Division Rings from Dimension Two to Three

2011 ◽  
Vol 18 (03) ◽  
pp. 437-446
Author(s):  
Gregor Dolinar ◽  
Janko Marovt
Keyword(s):  
Division Ring ◽  
Multiplicative Semigroup ◽  
Division Rings ◽  
Matrix Semigroups

Let 𝔻 be an arbitrary division ring and Mn(𝔻) the multiplicative semigroup of all n × n matrices over 𝔻. We describe the general form of non-degenerate homomorphisms from M2(𝔻) to M3(𝔻).

Subnormal Subgroups of Division Rings

1963 ◽  
Vol 15 ◽  
pp. 80-83 ◽  
Author(s):  
I. N. Herstein ◽  
W. R. Scott
Keyword(s):  
Normal Subgroup ◽  
Division Ring ◽  
Multiplicative Group ◽  
Finite Sequence ◽  
Division Rings ◽  
Subnormal Subgroups

Let K be a division ring. A subgroup H of the multiplicative group K′ of K is subnormal if there is a finite sequence (H = A0, A1, . . . , An = K′) of subgroups of K′ such that each Ai is a normal subgroup of Ai+1. It is known (2, 3) that if H is a subdivision ring of K such that H′ is subnormal in K′, then either H = K or H is in the centre Z(K) of K.

Derivations with Invertible Values on a Lie Ideal

1988 ◽  
Vol 31 (1) ◽  
pp. 103-110 ◽  
Author(s):  
Jeffrey Bergen ◽  
L. Carini
Keyword(s):  
Division Ring ◽  
Unit Element ◽  
Lie Ideal ◽  
Division Rings

AbstractLet R be a ring which possesses a unit element, a Lie ideal U ⊄ Z, and a derivation d such that d(U) ≠ 0 and d(u) is 0 or invertible, for all u ∈ U. We prove that R must be either a division ring D or D2, the 2 X 2 matrices over a division ring unless d is not inner, R is not semiprime, and either 2R or 3R is 0. We also examine for which division rings D, D2 can possess such a derivation and study when this derivation must be inner.

Open questions concerning antiautomorphisms of division rings with quasi-generalized Engel conditions

2019 ◽  
Vol 18 (09) ◽  
pp. 1950167 ◽  
Author(s):  
M. Chacron ◽  
T.-K. Lee
Keyword(s):  
Finite Order ◽  
Division Ring ◽  
Affirmative Answer ◽  
Major Result ◽  
Division Rings ◽  
Open Questions ◽  
Engel Condition ◽  
Open Question ◽  
Positive Integers

Let [Formula: see text] be a noncommutative division ring with center [Formula: see text], which is algebraic, that is, [Formula: see text] is an algebraic algebra over the field [Formula: see text]. Let [Formula: see text] be an antiautomorphism of [Formula: see text] such that (i) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers depending on [Formula: see text]. If, further, [Formula: see text] has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that [Formula: see text] is commuting, that is, [Formula: see text], all [Formula: see text]. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on [Formula: see text] can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring [Formula: see text] with an antiautomorphism [Formula: see text] satisfying the stronger condition (ii) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, [Formula: see text] has finite order then [Formula: see text] is commuting. We show here, that again the finite order assumption on [Formula: see text] can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

Relations in the semigroup of 2 × 2 upper-triangular matrices

2020 ◽  
Vol 30 (03) ◽  
pp. 567-584
Author(s):  
Henri-Alex Esbelin ◽  
Marin Gutan
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Semigroup Theory ◽  
Theoretical Computer Science ◽  
Multiplicative Semigroup ◽  
Triangular Matrices ◽  
Theoretical Computer ◽  
Upper Triangular Matrices ◽  
Matrix Semigroups

Let [Formula: see text] with [Formula: see text] be [Formula: see text] upper-triangular matrices with rational entries. In the multiplicative semigroup generated by these matrices, we check if there are relations of the form [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] We give algorithms to find relations of the previous form. Our results are extensions of some theorems obtained by Charlier and Honkala in [The freeness problem over matrix semigroups and bounded languages, Inf. Comput. 237 (2014) 243–256]. Our paper is at the interface between algebra, number theory and theoretical computer science. While the main results concern decidability and semigroup theory, the methods for obtaining them come from number theory.

scholarly journals Classical groups over division rings of characteristic two

1972 ◽  
Vol 7 (2) ◽  
pp. 191-226 ◽  
Author(s):  
William M. Pender ◽  
G.E. Wall
Keyword(s):  
Normal Subgroup ◽  
Quadratic Form ◽  
Division Ring ◽  
Quadratic Forms ◽  
Factor Group ◽  
Classical Groups ◽  
Isometry Groups ◽  
Division Rings ◽  
Characteristic Two

The notion of quadratic form over a field of characteristic two is extended to an arbitrary division ring of characteristic two with an involution of the first kind. The resulting isometry groups are shown to have a simple normal subgroup and the structure of the factor group is calculated. It is indicated how one may define and analyse all the classical groups in a unified manner by means of quadratic forms.

On a Theorem of Herstein

1969 ◽  
Vol 21 ◽  
pp. 1348-1353 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Division Ring ◽  
Monic Polynomial ◽  
Constant Term ◽  
Multiplicative Semigroup ◽  
Simple Ring ◽  
Left Hand ◽  
Ring Of Integers ◽  
Image Position

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that1then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:(1)*Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

scholarly journals Matrix semigroups over semirings

2019 ◽  
Vol 30 (02) ◽  
pp. 267-337
Author(s):  
Victoria Gould ◽  
Marianne Johnson ◽  
Munazza Naz
Keyword(s):  
Regular Ring ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Multiplicative Semigroup ◽  
Main Interest ◽  
Left Action ◽  
Triangular Matrices ◽  
Matrix Semigroup ◽  
Matrix Semigroups ◽  
Weakly Abundant

We study properties determined by idempotents in the following families of matrix semigroups over a semiring [Formula: see text]: the full matrix semigroup [Formula: see text], the semigroup [Formula: see text] consisting of upper triangular matrices, and the semigroup [Formula: see text] consisting of all unitriangular matrices. Il’in has shown that (for [Formula: see text]) the semigroup [Formula: see text] is regular if and only if [Formula: see text] is a regular ring. We show that [Formula: see text] is regular if and only if [Formula: see text] and the multiplicative semigroup of [Formula: see text] is regular. The notions of being abundant or Fountain (formerly, weakly abundant) are weaker than being regular but are also defined in terms of idempotents, namely, every class of certain equivalence relations must contain an idempotent. Each of [Formula: see text], [Formula: see text] and [Formula: see text] admits a natural anti-isomorphism allowing us to characterise abundance and Fountainicity in terms of the left action of idempotent matrices upon column spaces. In the case where the semiring is exact, we show that [Formula: see text] is abundant if and only if it is regular. Our main interest is in the case where [Formula: see text] is an idempotent semifield, our motivating example being that of the tropical semiring [Formula: see text]. We prove that certain subsemigroups of [Formula: see text], including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We also consider the subsemigroups [Formula: see text] and [Formula: see text] consisting of those matrices of [Formula: see text] and [Formula: see text] having all elements on and above the leading diagonal non-zero. We prove the idempotent generated subsemigroup of [Formula: see text] is [Formula: see text]. Further, [Formula: see text] and [Formula: see text] are families of Fountain semigroups with interesting and unusual properties. In particular, every [Formula: see text]-class and [Formula: see text]-class contains a unique idempotent, where [Formula: see text] and [Formula: see text] are the relations used to define Fountainicity, but yet the idempotents do not form a semilattice.

scholarly journals On subgroups in division rings of type 2

2012 ◽  
Vol 49 (4) ◽  
pp. 549-557
Author(s):  
Bui Hai ◽  
Trinh Deo ◽  
Mai Bien
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Division Rings ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Multiplicative Subgroups

Let D be a division ring with center F. We say that D is a division ring of type 2 if for every two elements x, y ∈ D, the division subring F(x, y) is a finite dimensional vector space over F. In this paper we investigate multiplicative subgroups in such a ring.

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