We classify infinitely generated projective modules over generalized Weyl algebras. For instance, we prove that over such algebras every projective module is a direct sum of finitely generated modules.

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely-generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely-generated Gorenstein-projective modules.