  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
$\sigma$-unital rings having a $\sigma$-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 $I_0$-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.
