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).

/pdf/infinite-abelian-groups-2dvcz0046a.pdf

Infinite Abelian Groups

The central concept of this thesis is that of Leavitt path algebras, a notion introduced by both Abrams and Aranda Pino in [AA1] and Ara, Moreno and Pardo in [AMP] in 2004. The idea of using a field K and row-finite graph E to generate an algebra LK(E) provides an algebraic analogue to Cuntz and Krieger’s work with C∗-algebras of the form C∗(E) (which, despite the name, are analytic concepts). At the same time, Leavitt path algebras also generalise the algebras constructed by W. G. Leavitt in [Le1] and [Le2], and it is from this connection that the Leavitt path algebras get their name. Although the concept of a Leavitt path algebra is relatively new, in the years since the publication of [AA1] there has been a flurry of activity on the subject. Many results were initially shown for row-finite graphs, then extended to countable (but not necessarily row-finite) graphs (as in [AA3]) and then finally shown for completely arbitrary graphs (see, for example, [AR]). Most of the research has focused on the connections between ring-theoretic properties of LK(E) and graphtheoretic properties of E (for example [AA2], [AR] and [ARM2]), the socle and socle series of a Leavitt path algebra ([AMMS1], [AMMS2] and [ARM1]) and analogues between LK(E) and their C ∗-algebraic equivalents C∗(E) (for example [To]). Some papers have classified certain sets of Leavitt path algebras, such as [AAMMS], which classifies the Leavitt path algebras of graphs with up to three vertices (and without parallel edges). In Chapter 1 we will cover the ring-, moduleand graph-theoretic background necessary to examine these algebras in depth, as well as taking a brief look at Morita equivalence, a concept that will prove useful at various points in this thesis.

/pdf/leavitt-path-algebras-584irhft0j.pdf

Leavitt Path Algebras

Almost free modules

theory of graph algebras

We show that if E is an arbitrary acyclic graph then the Leavitt path algebra LK(E) is locally K-matricial; that is, LK(E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graph-theoretic conditions (e.g. row-finiteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E: (1) LK(E) is von Neumann regular. (2) LK(E) is π-regular. (3) E is acyclic. (4) LK(E) is locally K-matricial. (5) LK(E) is strongly π-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.

Regularity Conditions for Arbitrary Leavitt Path Algebras

The theory of prime ideals of Leavitt path algebras over arbitrary graphs

Finitely presented simple modules over leavitt path algebras

We show that if $E$ is an arbitrary acyclic graph then the Leavitt path algebra $L_K(E)$ is locally $K$-matricial; that is, $L_K(E)$ is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field $K$. As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph $E$: (1) $L_K(E)$ is von Neumann regular. (2) $L_K(E)$ is $\pi$-regular. (3) $E$ is acyclic. (4) $L_K(E)$ is locally $K$-matricial. (5) $L_K(E)$ is strongly $\pi$-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.

Regularity conditions for arbitrary Leavitt path algebras

A torsion-free abelian group G is said to be a Butler group if Bext(C, T) = 0 for all torsion groups T. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group G satisfies the T.E.P. over a pure subgroup H if and only if H is decent in G in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called B2-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ^-group. We show under ( V = L) that this is indeed the case for Butler groups of rank Nt. On the other hand it is shown that, under ZFC, it is undecidable whether a group B for which Bext( B, T) = 0 for all countable torsion groups T is indeed a B2-group.

https://www.ams.org/tran/1988-305-01/S0002-9947-1988-0920150-X/S0002-9947-1988-0920150-X.pdf

Kulumani M. Rangaswamy

Papers

Regularity Conditions for Arbitrary Leavitt Path Algebras

The theory of prime ideals of Leavitt path algebras over arbitrary graphs

Finitely presented simple modules over leavitt path algebras

Regularity conditions for arbitrary Leavitt path algebras

Infinite rank Butler groups