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

Abstract

In this paper we prove that two idempotent rings are Morita equivalent if every corner of one of them is isomorphic to a corner of a matrix ring of the other one. We establish the converse (which is not true in general) for σ-unital rings having a σ-unit consisting of von Neumann regular elements. The following aim is to show that a property is Morita invariant if it is invariant under taking corners 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 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 infinite emitter does not admit any desingularization is also given.

GCR and CCR Steinberg Algebras

“…Recall that two rings R and S with local units are Morita equivalent if their categories of unitary modules are equivalent. We shall use a result of [23], which reduces the problem to checking corner invariance and invariance under matrix amplification (with finite index sets). This section will not be used in the rest of the paper and can be omitted.…”

“…A property of rings is said to be corner invariant [23] if R has the property if and only if all its corners eRe with e an idempotent have the property. We are now prepared to prove that CCR and GCR are corner invariant for rings with local units.…”

“…According to the main result of [23], to show that CCR and GCR are Morita invariant properties for rings with local units it suffices to show that they are corner invariant, which was done in Proposition 3.4, and that R has the property if and only if M n (R) has the property for every n ≥ 1. Proof.…”