  The study of invariants of group actions on commutative polynomial rings has
motivated many developments in commutative algebra and algebraic geometry. It
has been of particular interest to understand what conditions on the group
result in an invariant ring satisfying useful properties. In particular,
Watanabe's Theorem states that the invariant subring of $k[x_1,\ldots,x_n]$
under the natural action of a finite subgroup of $SL_n(k)$ is always
Gorenstein. In this paper, we study this question in the more general setting
of group actions on noncommutative non-connected algebras $A$. We develop the
notion of a homological determinant of an automorphism of $A$, then use the
homological determinant to study actions of finite groups $G$ on $A$. We give a
sufficient condition so that the invariant ring $A^G$ has finite injective
dimension and satisfies the generalized Gorenstein condition. More precisely,
let $A$ be a noetherian $\mathbb{N}$-graded generalized Gorenstein algebra with
finite global dimension. Suppose all elements of $G$ fix the idempotents of $A$
and act with trivial homological determinant. Then the invariant ring $A^G$ is
generalized Gorenstein.
