In this paper we prove that two idempotent rings are Morita equivalent ifevery corner of one of them is isomorphic to a corner of a matrix ring of theother one. We establish the converse (which is not true in general) for$sigma$-unital rings having a $sigma$-unit consisting of von Neumann regularelements. The following aim is to show that a property is Morita invariant ifit is invariant under taking corners and under taking matrices. The previousresults are used to check the Morita invariance of certain ring properties(being locally left/right artinian/noetherian, being categorically left/rightartinian, being an $I_0$-ring and being properly purely infinite) and certaingraph properties in the context of Leavitt path algebras (Condition (L),Condition (K) and cofinality). A different proof of the fact that a graph withan infinite emitter does not admit any desingularization is also given.
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