This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

In this chapter, we describe nine families of Moufang polygons. In Part III of this book, we prove that every Moufang polygon is isomorphic to one of the Moufang polygons described in this chapter.

An Inventory of Moufang Polygons

Abstract We develop the basic theory of a general class of symmetric spaces, called lineated symmetric spaces, that satisfy the axioms of Loos together with an additional axiom that guarantees unique midpoints of symmetry. Our primary interest is the case that these symmetric spaces are Banach manifolds, in which case they exhibit an interesting geometric structure, and particularly in the metric case, where it is assumed the symmetric space carries a convex metric, an invariant complete metric contracting the square root function. One major result is that the distance function between points evolving over time on two geodesics is a convex function. Primary examples arise from involutive Banach-Lie groups (G,σ) admitting a polar decomposition G = P · K, where K is the subgroup fixed by σ and P is the associated symmetric set. We consider an appropriate notion of seminegative curvature for such symmetric spaces endowed with an invariant Finsler metric and prove that the corresponding length metric must be a convex metric. The preceding results provide a general framework for the interesting Finsler geometry of the space of positive Hermitian elements of a C*-algebra that has emerged in recent years.

Symmetric spaces with convex metrics

Recently we have presented several structural results on certain isometries of spaces of positive definite matrices and on those of unitary groups. The aim of this paper is to put those previous results into a common perspective and extend them to the context of operator algebras, namely, to that of von Neumann factors.

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General Mazur–Ulam Type Theorems and Some Applications

We define generalized Clifford parallelisms in PG ( 3 , F ) with the help of a quaternion skew field H over a field F of arbitrary characteristic. Moreover we give a geometric description of such parallelisms involving hyperbolic quadrics in projective spaces over suitable quadratic extensions of F .

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Generalized Clifford parallelisms

Collineations of geometric structures derived from quaternion algebras

The correspondence between right loops (P, +) with the property “(*) ∀a, b ∈ P : a − (a − b) − b” and reflection structures described in [4] is extended to the class of graphs with parallelism (P, e, ∥). In this connection K-loops correspond with trapezium graphs, i.e. complete graphs with parallelism satisfying two axioms (T1), (T2) (cf. §3 ). Moreover (P, e, ∥ +) is a structure loop (i.e. for each a ∈ P the map a
 + : P → P; x → a + x is an automorphism of the graph with parallelism (P, e, ∥)) if and only if (P, +) is a K-loop or equivalently if (P, e, ∥) is a trapezium graph.

Loops, reflection structures and graphs with parallelism

We describe a representation of any semiregularleft loop by means of asemiregular bipartite involution set or, equivalently, a 1-factorization (i.e., a parallelism) of a bipartite graph, with at least one transitive vertex.

Left loops, bipartite graphs with parallelism and bipartite involution sets

This is a general frame for a theory which connects the areas of loops, involution sets and graphs with parallelism. Our main results are stated in §5, §6 and §7. In §5 we derive a partial binary operation from an involution set and we discuss if such operation is a Bol operation or a K-operation, in §6, we relate involution sets with loops. In §7 we look for the possibility to construct loop-nearrings by considering the automorphism groups of loops.

Silvia Pianta

Papers

Generalized Clifford parallelisms

Collineations of geometric structures derived from quaternion algebras

Loops, reflection structures and graphs with parallelism

Left loops, bipartite graphs with parallelism and bipartite involution sets

From Involution Sets, Graphs and Loops to Loop-Nearrings