  In order to study graded Frobenius algebras from a ring theoretical
perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings
and a shift-version of the latter ones, and we investigate the structure and
representations of such objects. We need to revisit graded simple graded left
Artinian rings, graded semisimple rings, and to provide graded versions of
certain results concerning the Jacobson radical, the singular radical, and
their connection to finiteness conditions and injectivity. We prove a structure
result for (shift-)graded Frobenius rings.
