  We develop the theory of groupoid graded semisimple rings. Our rings are
neither unital nor one-sided artinian. Instead, they exhibit a strong version
of having local units and being locally artinian, and we call them
$\Gamma_0$-artinian. One of our main results is a groupoid graded version of
the Wedderburn-Artin Theorem, where we characterize groupoid graded semisimple
rings as direct sums of graded simple $\Gamma_0$-artinian rings and we exhibit
the structure of this latter class of rings. In this direction, we also prove a
groupoid graded version of Jacobson-Chevalley density theorem. We need to
define and study properties of groupoid gradings on matrix rings (possibly of
infinite size) over groupoid graded rings, and specially over groupoid graded
division rings. Because of that, we study groupoid graded division rings and
their graded modules. We consider a natural notion of freeness for groupoid
graded modules that, when specialized to group graded rings, gives the usual
one, and show that for a groupoid graded division ring all graded modules are
free (in this sense). Contrary to the group graded case, there are groupoid
graded rings for which all graded modules are free according to our definition,
but they are not graded division rings. We exhibit an easy example of this kind
of rings and characterize such class among groupoid graded semisimple rings. We
also relate groupoid graded semisimple rings with the notion of semisimple
category defined by B. Mitchell. For that, we show the link between functors
from a preadditive category to abelian groups and graded modules over the
groupoid graded ring associated to this category, generalizing a result of P.
Gabriel. We characterize simple artinian categories and categories for which
every functor from them to abelian groups is free in the sense of B. Mitchell.
