Cohn-Leavitt Path Algebras and the Invariant Basis Number Property

TL;DR

This paper characterizes when separated Cohn-Leavitt path algebras of finite graphs possess the Invariant Basis Number property, linking algebraic structure with K-theory and Morita equivalence.

Contribution

It provides necessary and sufficient conditions for IBN in separated Cohn-Leavitt path algebras and explores their K-theoretic properties and Morita equivalences.

Findings

Separated Cohn path algebras always have IBN.

Non-stable K-theory of corner rings is determined by ambient rings.

Morita equivalent rings can be non-IBN and of different types.

Abstract

We give the necessary and sufficient condition for a separated Cohn-Leavitt path algebra of a finite digraph to have IBN. As a consequence, separated Cohn path algebras have IBN. We determine the non-stable K-theory of a corner ring in terms of the non-stable K-theory of the ambient ring. We give a necessary condition for a corner algebra of a separated Cohn-Leavitt path algebra of a finite graph to have IBN. We provide Morita equivalent rings which are non-IBN, but are of different types.