Definition 4

— In the notation of Prop. 9, those ideals $(p^n)$ ($p \in P, n \geq 1$ an integer) such that $m(p^n) > 0$ are called the elementary divisors of the module $M$, and the integers $m(p^n)$ are called their multiplicities; if the integer $m(0)$ is $> 0$, it is called the multiplicity of the elementary divisor 0.

As for the invariant factors (VII, p. 19, Def. 1), when $A = \mathbf{Z}$ or $A = K[X]$ (K a commutative field), then the canonical generator of the ideal $(p^n)$ (a positive integer or a monic polynomial) is also called, by abuse of language, an elementary divisor of the finitely generated module $M$.