Definition 1

— In the notation and hypotheses of Th. 1, the ideals $A\alpha_i$ of $A$ are called the invariant factors of the submodule $M$ with respect to the module $L$.

In the case where $A$ is either the ring $\mathbf{Z}$ of rational integers or the polynomial ring $K[X]$ in one indeterminate over a field $K$, there is a canonical way of choosing a generator for each ideal of $A$: a positive integer in the case of $\mathbf{Z}$, or a monic polynomial in the case of $K[X]$ (VII, p. 5). In each of these cases, the canonical generator of the invariant factor $Aa_i$ is also called an invariant factor of $M$ with respect to $L$, by abuse of language.