Definition 2

— In the notation and hypotheses of Th. 2, the ideals $a_k$ are called the invariant factors of the module E. As in Def. 1 (VII, p. 19), when $A = \mathbf{Z}$ or $A = K[X]$, the canonical generator of the ideal $a_k$ (a positive integer or a monic polynomial) is also called, by abuse of language, an invariant factor of the finitely generated module E.

We must be careful not to confuse the invariant factors of a module E with those of a submodule M of a free module L with respect to the module L (Def. I).