T. Y. Lam

Introduction to Quadratic Forms over Fields

Foundations Introduction to Witt rings Quaternion algebras and their norm forms The Brauer-Wall group Clifford algebras Local fields and global fields Quadratic forms under algebraic extensions Formally real fields, real-closed fields, and pythagorean fields Quadratic forms under transcendental extensions Pfister forms and function fields Field invariants Special topics in quadratic forms Special topics on invariants Bibliography Index.

A CRASH COURSE ON STABLE RANGE, CANCELLATION, SUBSTITUTION AND EXCHANGE

The themes of cancellation, internal cancellation, substitution and exchange have led to a lot of interesting research in the theory of modules over commutative and noncommutative rings. This article provides a quick and relatively self-contained introduction to the voluminous work in this area, using the notion of the stable range of rings as a unifying tool. With only a small number of exceptions, all theorems stated here are proved in full, modulo basic facts in the theory of modules and rings available in standard textbooks on ring theory.

Vandermonde and Wronskian matrices over division rings

In [L], the first author initiated the study of Vandermonde matrices with entries over a division ring K. Even in the case when K is a field, the definition of a Vandermonde matrix in [L] is already more general than that in the classical sense, as it takes into account a given endomorphism S of K. Under the epilogue of [L], it was further pointed out that the results of that paper remain valid if, in the definition of a Vandermonde matrix, one allows for a given S-derivation D and uses the appropriate definition of the “power functions” in the (S, D)-setting. Such a generalization is worthwhile because, in this setting, the study of Vandermonde matrices over K also encompasses the study of Wronskian matrices with respect to D, so results on Vandermonde matrices may be used to study linear differential equations arising from the operator D. More generally, Vandermonde matrices seem to be an important tool in studying the skew polynomial ring K[t, S, D], so a deeper understanding of Vandermonde matrices should be important to the study of the arithmetic of a division ring. The purpose of this paper is threefold. First, we shall develop (under Sec- tion 2) the basic facts on skew polynomials in the (S, D)-setting, define the evaluation of such polynomials, and prove the all-important “Product Theorem” (2.7). Second, we shall give a general computation of the rank of a Vandermonde matrix with respect to (S, D). As in [L], the computation is first reduced to the case when the elements a,, . . . . a, used to build the Vandermonde matrix are pairwise “(S, D)-conjugate.” In this case, the rank of the Vandermonde matrix is computed by the dimension of a vector space over a certain division subring of K (cf. Theorem 4.5). Last, under

Biquaternion algebras and quartic extensions

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Exercises in modules and rings

Free Modules.- Projective and Injective Modules.- Flat Modules and Homological Dimensions.- More Theory of Modules.- Rings of Quotients.- More Rings of Quotients.- Frobenius and Quasi-Frobenius Rings.- Matrix Rings, Categories of Modules, and Morita Theory.

Wedderburn polynomials over division rings, I

Abstract A Wedderburn polynomial over a division ring K is a minimal polynomial of an algebraic subset of K . Such a polynomial is always a product of linear factors over K , although not every product of linear polynomials is a Wedderburn polynomial. In this paper, we establish various properties and characterizations of Wedderburn polynomials over K , and show that these polynomials form a complete modular lattice that is dual to the lattice of full algebraic subsets of K . Throughout the paper, we work in the general setting of an Ore skew polynomial ring K [ t , S , D ], where S is an endomorphism of K and D is an S -derivation on K .

Algebraic Conjugacy Classes and Skew Polynomial Rings

The goal of this paper is to develop further the theory of skew polynomial rings over division rings, using as our main tools the notions of invariant and semi-invariant polynomials. These notions arise naturally when one tries to study the algebraic conjugacy classes (in a suitably generalized sense) of the underlying division ring. A substantial part of our effort will also be devoted to the investigation of the properties and the characterizations of algebraic derivations, algebraic endomorphisms, and their respective minimal polynomials. This investigation is made possible by the discovery of the relationship between polynomial equations and differential equations, and the relationship between polynomial dependence and linear dependence. Applications of these results to the study of non-commu tative Hilbert 90-type theorems will be presented in a forthcoming work [LL2].

Hilbert 90 theorems over division rings

Huberts Satz 90 is well-known for cyclic extensions of fields, but attempts at generalizations to the case of division rings have only been partly successful. Jacobsons criterion for logarithmic derivatives for fields equipped with derivations is formally an analogue of Satz 90, but the exact relationship between the two was apparently not known. In this paper, we study triples (K,S, D) where S is an endomorphism of the division ring K, and D is an S-derivation. Using the technique of Ore extensions K (t, S, D). we char- acterize the notion of (5, D)-algebraicity for elements a e K, and give an effective criterion for two elements a, b € K to be {S, £>)-conjugate, in the case when the (S, Z>)-conjugacy class of a is algebraic. This criterion amounts to a general Hubert 90 Theorem for division rings in the (K ,S, Z))-setting, sub- suming and extending all known forms of Hubert 90 in the literature, including the aforementioned Jacobson Criterion. Two of the working tools used in the paper, the Conjugation Theorem (2.2) and the Composite Function Theorem (2.3), are of independent interest in the theory of Ore extensions.

SUMS OF UNITS IN RINGS

In this paper, we study rings that are additively generated by units. We prove that if the identity in a ring with stable range one is a sum of two units, then every (von Neumann) regular element is a sum of two units. It follows that every element in a unit-regular ring is a sum of two units if the identity is a sum of two units. Also, if the identity of a strongly π-regular ring is a sum of two units, then every element is a sum of three units.

On uniform dimensions of ideals in right nonsingular rings

For any (S, R)-bimodule M, one can define an invariant d(M) by taking the supremum of n for which there exists a direct sum of nonzero subbimodules N = M1 ⊕ M2 ⊕ … ⊕ Mn such that N is essential in M as a right R-submodule. This invariant is a sort of hybrid between the right uniform dimension and the 2-sided uniform dimension. In this paper, we study the ideal structure of a right nonsingular ring R terms of the ideal structure of Qmaxr(R) by working with the invariant d(I) = d(RIR) for ideals I ⊂ R. The family F(R) of ideals I for which there exists an ideal J ⊂ R with I ⊕ J ⊂e Rr is characterized in various ways, and for I ∈ F(R), the invariant d(I) is related to the direct product decomposition of the ring E(IR) (injective hull) in Qmaxr(R). It is shown that d(I) is very well-behaved for the ideals I ∈ F(R) and various results are obtained on the relationship between d(I), u. dim(RIR) and u. dim(IR).