 Locally principal subvariety

A proper subvariety locally defined by one non-zero-divisor is an effective Cartier divisor. On an integral variety a nonzero local equation is automatically a non-zero-divisor. Its conormal sheaf is a line bundle, since $A/(f)\cong (f)/(f^2)$ via multiplication by $f$. Allowing the zero equation would include the whole variety and would not give this conclusion.