In Liu's Algebraic Geometry and Arithmetic Curves, Definition 7.1.17, a few lines after the definition of principal Cartier divisor (as one in the image of $\Gamma (X,K_X^*) \to \Gamma (X,K_X^*/ \mathcal{O} _X^*)$), we read (for an arbitrary scheme $X$):
[A Cartier divisor] is principal if it can be represented by a system {$(X,f)$}.
I understand that this is only an "if" and not an "if and only if" in general. Am I right? (if $X$ is integral or affine noetherian we have an "only if").