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Most of the rings one encounters as "basic
examples" have what's known as the "Invariant Basis Number" property,
namely, for every pair of positive integers m and n, if
the free left R-modules RR(m) and
RR(n) are isomorphic, then m=n.
There are, however, large classes of rings which do not have this
property. While at first glance such rings might seem pathological, in
fact they arise quite naturally in a number of contexts (e.g. as
endomorphism rings of infinite dimensional vector spaces), and possess
a significant (perhaps surprising) amount of structure. We describe a
class of such rings, the (now-classical) Leavitt algebras, and
then describe their recently developed generalizations, the Leavitt
path algebras. One of the nice aspects of this subject is that
pictorial representations (using graphs) of the algebras are readily
available. In addition, there are strong connections between these
algebraic structures and a class of C*-algebras, a connection which is
currently the subject of great interest to both algebraists and
analysts. |
