Let $K \subseteq L$ be number fields and $S$ be an order in $L$ (not necessarily maximal). Let $R:=S\cap K$ and $S^*$ such as $R^*$ denote the trace duals of $S$ and $R$, respectively. Then $S^*$ and $R^*$ are fractional ideals of $S$ and $R$, respectively.
I am looking for a sufficient condition on $S$ such that
(1) $R^*$ is invertible in $R$ and
(2) $R^*S$ is invertible in $S$.
Let me note that (1) is an equivalent description of $R$ being Gorenstein, so my first approach was to assume $S$ to be Gorenstein, but i was not able to deduce (1) and (2) from this.
Thank you for every hint!