Leavitt path algebras and the IBN property

Abstract

A ring has invariant basis number property (IBN) if any two bases of a finitely generated free module have the same number of elements. In 1960's Leavitt constructed examples of rings R without IBN, more precisely for any positive integers m < n the ring R has a free module with a basis of m elements and another basis with n elements but no bases with k elements if k < n and not equal to m. Now, R is called a Leavitt algebra of type (m; n) and denoted by L(m; n). The Leavitt path algebras were defined just over a decade ago but they have roots in the works of Leavitt, as L(1; n) is algebra isomorphic to the Leavitt path algebra of the graph of a rose with n petals. Also, Cohn-Leavitt path algebras are a generalization of Leavitt path algebras which has both IBN and non-IBN examples. We will give the necessary and sufficient condition for a Cohn-Leavitt path algebra of a finite graph to have IBN. By using the non-stable K-theory, we provide Morita equivalent rings which are non-IBN, but are of different types. (This is joint work with M.Ozaydin).