  We extend the classical construction by Noether of crossed product algebras,
defined by finite Galois field extensions, to cover the case of separable (but
not necessarily finite or normal) field extensions. This leads us naturally to
consider non-unital groupoid graded rings of a particular type that we call
object unital. We determine when such rings are strongly graded, crossed
products, skew groupoid rings and twisted groupoid rings. We also obtain
necessary and sufficient criteria for when object unital groupoid graded rings
are separable over their principal component, thereby generalizing previous
results from the unital case to a non-unital situation.
