Fully group graded algebras and a theorem of Fong

It is proved that any algebra fully graded by a finite group over a complete discrete valuation ring with an algebraically closed residue field of characteristic a prime p is Morita equivalent to an embedded graded subalgebra which is a crossed product; and an explicit way to get a decomposition of unity with a bounded length is shown. When the finite group is p-solvable, a theorem of Fongʹs type for fully graded algebras is obtained.