Dimension of a fractional ideal at residue fields (awaiting review)

Let $R$ be an order in an étale algebra $K$ and $I$ a fractional $R$-ideal. For every prime $\mathfrak{P}$ of $R$, the quotient $I/\mathfrak{P} I$ is a finite dimensional vector space over the finite field $R/\mathfrak{P}$. This dimension is $1$ if and only if the localization $I_\mathfrak{P}$ of $I$ at $\mathfrak{P}$ is a principal $R_\mathfrak{P}$-ideal. The ideal $I$ is invertible if and only if it is locally principal at every prime of $R$.

If the ideal $I$ is not locally principal at $\mathfrak{P}$ then $\mathfrak{P}$ is a singular prime of $R$.

If $R$ is Gorenstein then every fractional $R$-ideal $I$ with multiplicator ring $(I:I)=R$ is invertible (as an $R$-ideal).

For each fractional $R$-ideal $I$ with $(I:I)=R$, we record the dimensions of $I/\mathfrak{P}I$ over $R/\mathfrak{P}$ where $\mathfrak{P}$ runs over the singular primes of $R$.