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.

scholarly journals On the dimension of $H^{*}((\mathbb Z_2)^{\times t}, \mathbb Z_2)$ as a module over Steenrod ring

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Positive Integer ◽  
Tensor Product ◽  
Finite Field ◽  
Vector Space ◽  
Linear Group ◽  
Computer Algebra ◽  
General Linear Group ◽  
Polynomial Algebra ◽  
Minimal Set ◽  
Trivial Subgroup

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by assigning degree $1$ to each generator. We are interested in determining a minimal set of generators for the ring of invariants $(\mathbb P^{\otimes t})^{G_t}$ as a module over Steenrod ring, $\mathscr A_2.$ Here $G_t$ is a subgroup of the general linear group $GL(t, \mathbb Z_2).$ Equivalently, we want to find a basis of the $\mathbb Z_2$-vector space $\mathbb Z_2\otimes_{\mathscr A_2} (\mathbb P^{\otimes t})^{G_t}$ in each degree $n\geq 0.$ The problem is proved surprisingly difficult and has been not yet known for $t\geq 5.$ In the present paper, we consider the trivial subgroup $G_t = \{e\}$ for $t \in \{5, 6\},$ and obtain some new results on $\mathscr A_2$-generators for $(\mathbb P^{\otimes 5})^{G_5}$ in degree $5(2^{1} - 1) + 13.2^{1}$ and for $(\mathbb P^{\otimes 6})^{G_6}$ in "generic" degree $n = 5(2^{d+4}-1) + 47.2^{d+4}$ with a positive integer $d.$ An efficient approach to studying $(\mathbb P^{\otimes 5})^{G_5}$ in this case has been provided. In addition, we introduce an algorithm on the MAGMA computer algebra for the calculation of this space. This study is a continuation of our recent works in \cite{D.P2, D.P4}.

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.

scholarly journals Abelian Unipotent Subgroups of Finite Unitary and Symplectic Groups

1982 ◽  
Vol 33 (3) ◽  
pp. 331-344 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Unitary Group ◽  
Symplectic Group ◽  
General Linear ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Abelian Subgroups ◽  
Symplectic Case

AbstractIf G is the unitary group U(V) or the symplectic group Sp(V) of a vector space V over a finite field of characteristic p, and r is a positive integer, we determine the abelian p-subgroups of largest order in G whose fixed subspaces in V have dimension at least r, with the restriction that we assume p ≠ 2 in the symplectic case. In particular, we determine the abelian subgroups of largest order in a Sylow p-subgroup of G. Our results complement earlier work on general linear and orthogonal groups.

Endoprime modules and their direct sums

2018 ◽  
Vol 17 (08) ◽  
pp. 1850155 ◽  
Author(s):  
Gangyong Lee ◽  
S. Tariq Rizvi
Keyword(s):  
Direct Summand ◽  
Prime Ring ◽  
Endomorphism Ring ◽  
Matrix Ring ◽  
Direct Sums ◽  
Direct Sum ◽  
Finite Matrix ◽  
Reduced Ring ◽  
Baer Modules ◽  
Special Classes

The purpose of this paper is to further study the endoprime modules as one of the special classes of quasi-Baer modules. As a module theoretic analogue of a prime ring, we characterize an endoprime module via its endomorphism ring and a weak retractability condition. It is shown that any direct summand of an endoprime module is an endoprime module. A characterization is obtained when a direct sum of endoprime modules is an endoprime module. It is well known that every prime ring is semicentral reduced. We prove that a column (and row) finite matrix ring over a semicentral reduced ring is also a semicentral reduced ring. Consequently, it is shown that a column (and row) finite matrix ring over a prime ring is prime. Applications and examples illustrating our results are provided.

scholarly journals Abelian 2-subgroups of finite symplectic groups in characteristic 2

1982 ◽  
Vol 33 (3) ◽  
pp. 345-350 ◽  
Author(s):  
M. J. J. Barry ◽  
W. J. Wong
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Vector Space ◽  
Symplectic Group ◽  
Preceding Paper ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Characteristic 2 ◽  
Finite Symplectic Groups

AbstractIf Sp(V) is the symplectic group of a vector space V over a finite field of characteristic p, and r is a positive integer, the abelian p-subgroups of largest order in Sp(V) whose fixed subspaces in V have dimension at least r were determined in the preceding paper, in the case p ≠ 2. Here we deal with the case p = 2. Our results also complete earlier work on the orthogonal groups.

Generalized Derivations with Periodic Values

2015 ◽  
Vol 22 (01) ◽  
pp. 163-168
Author(s):  
Kun-Shan Liu
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Matrix Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Generalized Derivations ◽  
Fixed Positive Integer

Let R be a prime ring and n > 1 be a fixed positive integer. If g is a nonzero generalized derivation of R such that g(x)n=g(x) for all x ∈ R, then R is commutative except when R is a subring of the 2 × 2 matrix ring over a field. Moreover, we generalize the result to the case g(f(xi))n = g(f(xi)) for all x1, x2, …, xt∈ R, where f(Xi) is a multilinear polynomial.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

scholarly journals THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 418-423 ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Bilinear Form ◽  
Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

scholarly journals Weakly one-based geometric theories

2012 ◽  
Vol 77 (2) ◽  
pp. 392-422 ◽  
Author(s):  
Alexander Berenstein ◽  
Evgueni Vassiliev
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Geometric Theory ◽  
Common Generalization

AbstractWe study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over a finite field.

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