Let $R$ be a ring and $f \in R[[x]]$. Do we know anything about the ring $R[[x]]/(f)$. I understand that by making $R$ local and complete and adding some requirements on $f$, we can say a lot by using the Weierstrass preparation theorem.
Is there anything we can say in the more general case? I would be particularly interested in the rank of $R[[x]]/(f)$ over $R$, since I am not even sure it should be finite. Is it easier if $f \in R[x] \subset R[[x]]$?