Morita Equivalence and Morita Invariant Properties: Applications in the Context of Leavitt Path Algebras

Abstract

In this paper we prove that if two idempotent rings R and S are Morita equivalent then for every von Neumann regular element a ∈ R the local algebra of R at a, R a , is isomorphic to \(\mathbb{M}_{n}(S)_{u}\) for some natural n and some idempotent u in \(\mathbb{M}_{n}(S)\). We give examples showing that the converse of this result is not true in general and establish the converse for σ-unital rings having a σ-unit consisting of von Neumann regular elements.Our next aim is to prove that, for idempotent rings, a property is Morita invariant if it is invariant under taking local algebras at von Neumann regular elements and under taking matrices.The previous results are used to check the Morita invariance of certain ring properties (being locally left/right artinian/noetherian, being categorically left/right artinian, being an I0-ring and being properly purely infinite) and of certain graph properties in the context of Leavitt path algebras (Condition (L), Condition (K) and cofinality). A different proof of the fact that a graph with an uncountable emitter does not admit a desingularization is also given.