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).

We investigate Lie derivations on a class of algebras called triangular algebras. In particular, we give sufficient conditions such that every Lie derivation on such an algebra is a sum of derivation on and a mapping from to its centre.

Lie Derivations of Triangular Algebras

We give a method to obtain the primitive central idempotent of the rational group algebra ℚG over a finite group G associated to a monomial irreducible character which does not involve computations with the character field nor its Galois group. We also show that for abelian-by-supersolvable groups this method takes a particularly easy form that can be used to compute the Wedderburn decomposition of ℚG.

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On Monomial Characters and Central Idempotents of Rational Group Algebras

Biderivations of triangular algebras

Automorphisms and derivations of upper triangular matrix rings

Derivations of Upper Triangular Matrix Rings

Let FG be the group algebra of a group G over a field F and let * denote the canonical involution of FG induced by the map g→g

 −1
 ,gG. Let Un(FG)={uFG|uu

 *
 =1} be the group of unitary units of FG. In case char F=0, we classify the torsion groups G for which Un(FG) satisfies a group identity not vanishing on 2-elements. Along the way we actually prove that, in characteristic 0, the unitary group Un(FG) does not contain a free group of rank 2 if FG

 −
 , the Lie algebra of skew elements of FG, is Lie nilpotent. Motivated by this connection we characterize most groups G for which FG

 −
 is Lie nilpotent and char F≠2.

Unitary units and skew elements in group algebras

Star-polynomial identities: Computing the exponential growth of the codimensions ☆

We consider the isomorphism problem for partial group rings and show that, in the modular case, if and $R_{\hbox{\scriptsize\it par}}G_1\,{\cong}\, R_{\hbox{\scriptsize\it par}}G_2$ then the corresponding group rings of the Sylow -subgroups are isomorphic. We use this to prove that finite abelian groups having isomorphic modular partial group algebras are isomorphic. Moreover, in the integral case, we show that the isomorphism of partial group rings of finite groups and implies .

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Isomorphisms of partial group rings

In this paper, we consider all metacyclic groups of the type 〈a,b | an - 1, b 2 = 1, ba = aib〉 and give a concrete description of their rational group algebras. As a consequence we obtain, in a natural way, units which generate a subgroup of finite index in the full unit group, for almost all such groups.

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C. Polcino Milies

Papers

Derivations of Upper Triangular Matrix Rings

Unitary units and skew elements in group algebras

Star-polynomial identities: Computing the exponential growth of the codimensions ☆

Isomorphisms of partial group rings

Units of integral group rings of some metacyclic groups