For a global function field $K$, What is the difference between the ray class group modulo $\mathfrak{m}$ (as defined in say, JS Milne's CFT notes) and the $S$-class group (as defined in say, Rosen's book "Number Theory in Function Fields") for $S$ being equal to the support of modulus $\mathfrak{m}$ ?
the ray class group is $I(\mathfrak{m})/P(\mathfrak{m})$, where $I(\mathfrak{m})$ is the free abelian group generated by primes which do not divide $\mathfrak{m}$ (sounds like the group of $S$-divisors). And $P(\mathfrak{m})$ is the set of principle divisors of $\{a \in K^* : v_p(x -1 ) \geq \mathfrak{m}_p \}$ (I am ignoring real places because $K$ is a function field). This looks a lot like the principle $S$ divisors to me.
Rosen states that the S-class group is finite, whereas the notes from Pete Clark at http://alpha.math.uga.edu/~pete/8410Chapter8.pdf suggests that the ray class group mod $\mathfrak{m}$ is only finite if the constant field of $K$ is algebraically closed. I note that Clark does not require that $\mathfrak{m}_p > 0$, could this be the cause of the discrepancy, or am I otherwise confused ?