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We describe "quasi canonical modules" for modular invariant rings $R$ offinite group actions on factorial Gorenstein domains. From this we derive ageneral "quasi Gorenstein criterion" in terms of certain 1-cocycles. Thisgeneralizes a recent result of A. Braun for linear group actions on polynomialrings, which itself generalizes a classical result of Watanabe for non-modularinvariant rings.  We use an explicit classification of all reflexive rank one $R$-modules,which is given in terms of the class group of $R$, or in terms of$R$-semi-invariants. This result is implicitly contained in a paper of Nakajima(\cite{Nakajima:rel_inv}).

Relative Invariants, Ideal Classes and Quasi-Canonical Modules of Modular Rings of Invariants