Certain functional identities on division rings of characteristic two

Let R be a primitive ring with involution *. Thus R may be considered as an irreducible ring of endomorphisms of an additive abelian group V, so that D=HomR(V, V) is a division ring. Let C be the center of D. We shall furthermore assume that CRCR. It can be shown that an involution -y--+is induced in C which has the property that 1yx = (tyx*)* for all xER. The involution * is of the first kind if -y-+-7 is the identity mapping and is of the second kind if there is a -y Oe C such that fy = -y. The set of symmetric elements of R will be denoted by S. We now assume that S satisfies a nontrivial generalized polynomial identity over C (in the sense of Amitsur). This means that there exists a nonzero element f(xl, x2, * * *, x.) in the so-called C-universal product R(x) of the C-algebra R and the free C-algebra C[X,, x2,* * *, x,, ... ] in noncommuting indeterminants xi, X2, *. * Xn, ...* such thatf(s1, S2, * * *, sn) = 0 for all SI S2, *.*.*, SnCS. For more precise details concerning the above notions we refer the reader to [1, ?4], and [2, ?3]. The usual linearization process may be used so that we may assume without loss of generality that S satisfies a nontrivial generalized (homogeneous) multilinear identity of degree n in xi, x2, ,n:

Structural Properties of Models of N1-Categorical Theories

Motivated by the generalized polynomial identities in [1] and [5], our aim is to generalize some of the results in these works. Precisely, we extend the result in [1] on commuting values of the same generalized derivations to the different generalized derivations case by a short proof. Also as an application, we extend a result in [5] on images of a linear map with derivations to generalized derivations case.

Let be a division ring which is not a field with characteristic different from 2 and 3, and with n > 1. The purpose of this paper is to characterize additive maps f,g : satisfying the functional identity f(X) + X 2 g(X −1) = 0 for all invertible .

This paper presents some new rational functional identities on a division ring [Formula: see text] of characteristic not equal to [Formula: see text]. We prove that if the additive maps [Formula: see text] satisfy [Formula: see text], then both [Formula: see text] and [Formula: see text] are of the form [Formula: see text] where [Formula: see text] is an additive map and [Formula: see text] is a fixed element of [Formula: see text]. Further, we show that the only additive maps [Formula: see text] over [Formula: see text] satisfying [Formula: see text] where [Formula: see text] are the zero maps.

Let K be a field. We obtain conditions for elements of the form {1 + a i, 1 + b j} to generate a free group of rank 2 in a quaternion algebra over K, including the case where the characteristic of the field K is 2, based on an article by Professors Jairo Gonçalves, Arnaldo Mandel and Mazi Shirvani. These results are then applied to obtain pairs of elements that freely generate a free group in many other classes of division rings, such as total fields of fractions of Ore domains and Malcev-Neumann series rings. With these results, we partially answer a conjecture by A. Lichtman, regarding the existence of non-abelian free groups in the multiplicative group of non-commutative division rings.

Let A be an algebra with 1 over a field F and let B be a fixed F-basis of A. Let F〈x〉=F〈x1,…, xn,…,〉 be the free algebra over F in noncommutative indeterminates x1,…, xn,…, and denote by AF〈x〉 the free product of A and F〈x〉 over F. The elements of AF〈x〉 of the form varies, repetitions allowed) form an F-basis of AF〈x〉. They will be referred to as basis monomials, and the involved in a particular basis monomial will be called the coefficients of that basis monomial.

This thesis studies the conditions under which certain U (G)-modules have integral dimension. Specifically, the paper explores the circumstances under which certain U (G)-modules are free modules. Two main theorems are proven for this endeavor. The first theorem takes the matrix representation A of a submodule N of a free left module M over an arbitrary ring R. The theorem states that if A can be put into reduced row echelon form, then N and M/N are both free left R-modules. The second theorem takes the matrix representation B of an arbitrary left R-module homomorphism T. If B can be put into reduced column echelon form, then Ker( T) is a free left R-module as well. The results of these theorems are well-known if R is a field or division ring. The results are also probably known for an arbitrary ring R , but there does not seem to be a readily accessible source in the standard literature.

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients can be factorized on linear factors with coefficients from the same field. However, this is not true for numerous other equally important and interesting fields such as, for example, finite fields. All existing finite fields are of the q=pk order, where p is a prime number and k is a positive integer. Any two finite fields of the same order are mutually isomorfic, therefore the field of q order is unique; it is also known as »the Galois field«, named after the French mathematician Evariste Galois (1811-1832). This thesis deals with factorization of polynomials over finite fields. The general rule states that some polynomials are not reducible, yet each polynomial with coefficients of the field can be uniquely factorized into irreducible factors. There are various known algorithms that can determine whether or not a polynomial is reducible and – in the case of reducible polynomials – also find this factorization. There are several such algorithms, however, this thesis shall focus on the Berlekamp's algorithm. The introductory chapter introduces the historical development of the various approaches to polynomial factorization and presents some modern applications that are currently in use. The second chapter focuses on basic concepts and characteristics of algebraic structures and some classic results that are crucial for the understanding of the algorithm (e.g. groups, rings, fields, division rings and polynomial rings.). The algorithm also uses the Chinese remainder theorem for polynomials and Euclidean algorithm for calculating the greatest common divisor of two polynomials. The first part of Chapter 3 (i.e. the main chapter of the thesis) demonstrates 4 naive methods for the finding of polynomial factorization. For each of the methods we first introduced factorization of a polynomial over real numbers and then continued with an example of a polynomial with coefficients in the finite field. The sieve method, for instance, builds a list of irreducible polynomials of low degrees. Irreducible polynomials up to (and including) degree 6 over the finite fields Z2, Z3, Z5 and Z7 can be calculated using our own code made in the VisualC# programming language. These naive methods are followed by a detailed description of the Berlekamp's algorithm with theorems, proofs and various examples. At the end of section we present our implementation of the algorithm in the computational algebra system MAGMA.

Let $D$ be a division ring with infinite center $F$; $σ$ be an anti-automorphism of $D$ and $m$ be a positive integer such that $σ^m\neq \mathrm{Id}$. In this paper, we show that if $D$ satisfies a $σ^m$-GRI, then $D$ is centrally finite.

LetA be a prime ring with maximal right ring of quotients Q and with extended centroid C. Further, let S be a set, let αS→ A be a map, let m be a positive integer and let Ei Fi S m-1, be maps. We study functional identities of the form for all s 1,s 2,⋯,sm ϵ S (where , etc). If Sα is an (m+ l)-free subset of Q (for example, if S α = A and A is not algebraic of bounded degree m+1 over C, or S α is a ncentral Lie ideal of A and A is not algebraic of bounded degree m + 2 over C), definitive results are obtained.

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form nα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^\\alpha $$\\end{document}. We prove the inequality when α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is an even natural number with the sharp constant and remainder terms. We also find explicit constants in standard and weighted Rellich inequalities(with weights nα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^\\alpha $$\\end{document}) which are asymptotically sharp as α→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow \\infty $$\\end{document}. As a by-product of this work we derive a combinatorial identity using purely analytic methods, which suggests a plausible correlation between combinatorial and functional identities.

<abstract><p>A Prudnikov sum is extended to derive the finite sum of the Hurwitz-Lerch Zeta function in terms of the Hurwitz-Lerch Zeta function. This formula is then used to evaluate a number trigonometric sums and products in terms of other trigonometric functions. These sums and products are taken over positive integers which can be simplified in certain circumstances. The results obtained include generalizations of linear combinations of the Hurwitz-Lerch Zeta functions and involving powers of 2 evaluated in terms of sums of Hurwitz-Lerch Zeta functions. Some of these derivations are in the form of a new recurrence identity and finite products of trigonometric functions.</p></abstract>

Let 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.

A polynomial in noncommuting variables is vanishing, nil or central in a ring, R R , if its value under every substitution from R R is 0, nilpotent or a central element of R R , respectively. THEOREM. If R R has no nonvanishing multilinear nil polynomials then neither has the matrix ring R n {R_n} . THEOREM. Let R R be a ring satisfying a polynomial identity modulo its nil radical N N , and let f f be a multilinear polynomial. If f f is nil in R R then f f is vanishing in R / N R/N . Applied to the polynomial x y − y x xy - yx , this establishes the validity of a conjecture of Herstein’s, in the presence of polynomial identity. THEOREM. Let m m be a positive integer and let F F be a field containing no m m th roots of unity other than 1. If f f is a multilinear polynomial such that for some n > 2 f m n > 2{f^m} is central in F n {F_n} , then f f is central in F n {F_n} . This is related to the (non)existence of noncrossed products among p 2 {p^2} -dimensional central division rings.