Definition 1

The integer $e_q(M)$ is called the multiplicity of the A-module M relative to the ideal q.

It is also denoted by $e_q^A(M)$ when it is desired to mention the ring A. When A is local with maximal ideal m, one writes $e(M)$ or $e^A(M)$ for $e_m(M)$ or $e_m^A(M)$.