Hostname: page-component-6b989bf9dc-pmhlf Total loading time: 0 Render date: 2024-04-13T03:39:31.787Z Has data issue: false hasContentIssue false

Representations of algebras as universal localizations

Published online by Cambridge University Press:  15 January 2004

Center for Mathematics and its Applications, School of Mathematical Sciences, John Dedman Building, The Australian National University, Canberra ACT 0200, Australia. e-mail:
School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Edinburgh EH9 3JZ, Scotland. e-mail:
School of Mathematics, University of Bristol, Bristol BS8 1TW. e-mail:


Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra $S$ in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adjoin inverses to some of the arrows in the quiver, we show that the path algebra of the new quiver with relations is $M_n(S)$ where $n$ is the number of vertices in our quiver. The slogan would be that every finitely presented algebra is Morita equivalent to a universal localization of a finite dimensional algebra.

Research Article
2004 Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)