Let $G$ be a group. Below, the author talks about strictly $\Bbb{Z}$-group rings.
Definition 1.1.3.
(i) The augmentation map is the homomorphism $\varepsilon : \Bbb{Z}[G] \to \Bbb{Z}$ given by $$ \varepsilon(\sum\limits_{g \in G} a_g g) = \sum\limits_{g \in G} a_g. $$ (ii) The augmentation ideal $I_G$ is the kernel of the augmentation map $\varepsilon$.
It's intuitively clear to me that this is a $\Bbb{Z}$-module homomorphism, and not a ring homomorphism. Thus any kernel is therefore a $\Bbb{Z}$-submodule. So why here is it called an ideal?