On strongly graded Gorestein orders

Let $G$ be a finite group and let $\Lambda=\oplus_{g\in G}\Lambda_{g}$ be a strongly $G$-graded $R$-algebra, where $R$ is a commutative ring with unity. We prove that if $R$ is a Dedekind domain with quotient field $K$, $\Lambda$ is an $R$-order in a separable $K$-algebra such that the algebra $\Lambda_1$ is a Gorenstein $R$-order, then $\Lambda$ is also a Gorenstein $R$-order. Moreover, we prove that the induction functor $ind:Mod\Lambda_{H}\rightarrowMod\Lambda$ defined in Section 3, for a subgroup $H$ of $G$, commutes with the standard duality functor.