Example

(1) Let $\phi$ be a homomorphism of a ring A into a ring B; the mapping $(a, x) \mapsto \phi(a)x$ (resp. $(a, x) \mapsto x\phi(a)$) of $A \times B$ into B defines on B a left (resp. right) A-module structure. In particular if $\phi$ is taken to be the identity mapping on A, a canonical left (resp. right) A-module structure is obtained on A; to avoid confusion, the set A with this structure is denoted by $A_s$ (resp. $A_d$).

(2) On a commutative group G (written additively) the group with operators structure defined by the external law $(n, x) \mapsto n.x$ (I, § 3, no. 1) is a module structure over the ring $\mathbf{Z}$ of rational integers.

(3) Let E be a commutative group written additively, $\mathscr{E}$ the endomorphism ring of E (I, § 8, no. 3: recall that the product $fg$ of two endomorphisms is by definition the composite endomorphism $f \circ g$). The external law $(f, x) \mapsto f(x)$ between operators $f \in \mathscr{E}$ and elements $x \in E$ defines on E a canonical left $\mathscr{E}$-module structure.

Consider now a ring A and suppose there is given on E a left (resp. right) A-module structure; for all $\alpha \in A$, the homothety $h_\alpha : x \mapsto \alpha x$ (resp. $x \mapsto x\alpha$) belongs to $\mathscr{E}$; the mapping $\phi : \alpha \mapsto h_\alpha$ is a homomorphism of the ring A (resp. the opposite ring $A^0$) into the ring $\mathscr{E}$ and by definition $\alpha x = (\phi(\alpha))(x)$ (resp. $x\alpha = (\phi(\alpha))(x)$). Conversely, giving a ring homomorphism $\phi : A \to \mathscr{E}$ (resp. $\phi : A^0 \to \mathscr{E}$) defines a left (resp. right) A-module structure on E by the above formulae. In other words, being given a left (resp. right) A-module structure on an additive group E with additive law the given group law is equivalent to being given a ring homomorphism $A \to \mathscr{E}$ (resp. $A^0 \to \mathscr{E}$).