$\textbf{Q:}$ What is the relationship between conductors in class field theory and conductors in ring theory context? Conductors in class field theory context measures ramification.(However, I do not see why this should coincide with the notion of conductors in ring theory context.) Conductor in ring extension $A\subset B$ is $A:B$ where $B$ is treated as $A-$module. Then $A:B$ certainly measures deviation of $B$ to $A$ and most of time measures integral closureness. Is this merely coincidence of naming or there is connection between the two notions?
Ref. Nagata, Local Rings Chpt 1, Sec 10.