Cohen-Macaulay type (awaiting review)

The Cohen-Macaulay type of an order $S$, is the minimal number of generators (as an $S$-module) of a canonical module of $S$. Note that the trace dual ideal $S^t =\{ a : \mathrm{Tr}(aS) \subseteq \mathbb{Z} \}$ is a canonical module for $S$. In particular, the Cohen-Macaulay type equals $\max \{ \dim_{S/\mathfrak{l}} \frac{S^t}{\mathfrak{l}S^t} \}_\mathfrak{l}$, where $\mathfrak{l}$ runs over the maximal ideals of $S$. The order is called Gorenstein if the Cohen-Macaulay type equals $1$. This is the case precisely when every fractional $S$-ideal $I$ with $(I:I)$ is invertible.