Example

It has been seen (no. 1) that a homothety $h_\alpha : x \mapsto \alpha x$ on an $A$-module $E$ is not necessarily a linear mapping. But if $\alpha$ is invertible, $h_\alpha$ is a semi-linear mapping (which is moreover bijective) relative to the inner automorphism $\xi \mapsto \alpha \xi \alpha^{-1}$ of $A$, for $\alpha(\lambda x) = (\alpha \lambda \alpha^{-1})(\alpha x)$.

Let C be a third ring, $\rho': C \to B$ a homomorphism and G a C-module. If $v: G \to F$ is a semi-linear mapping relative to $\rho'$, the composition $w = u \circ v$ is a semi-linear mapping of G into E relative to the homomorphism $\rho'' = \rho \circ \rho'$. If $\rho$ is an isomorphism and $u: F \to E$ is a bijective semi-linear mapping relative to $\rho$, the inverse mapping $u': E \to F$ is a semi-linear mapping relative to the inverse isomorphism $\rho': A \to B$ of $\rho$.

It is thus seen that, for the species of structure defined by giving on an ordered pair (A, E) of sets a ring structure on A and a left A-module structure on E, the dimorphisms $(u, \phi)$ can be taken as morphisms (Set Theory, IV, § 2, no. 1); we shall always assume in what follows that this choice of morphisms has been made.

Remark (3). Let $A_1, A_2$ be two rings, $A = A_1 \times A_2$ their product and let $e_1 = (1, 0),\ e_2 = (0, 1)$ in A, so that $A_1$ and $A_2$ are canonically identified with the two-sided ideals $Ae_1$ and $Ae_2$ of A. For every A-module E, $e_1E$ and $e_2E$ are sub-A-modules $E_1,\ E_2$ of E, annihilated respectively by $e_2$ and $e_1$, so that, canonically identifying $A/Ae_2$ with $A_1$ and $A/Ae_1$ with $A_2$, $E_1$ (resp. $E_2$) is given an $A_1$-module (resp. $A_2$-module) structure. Moreover, E is the direct sum of $E_1$ and $E_2$, for every $x \in E$ can be written as $x = e_1x + e_2x$ and the relation $e_1x = e_2y$ implies $e_1x = e_1^2x = e_1e_2y = 0$. Conversely, for every ordered pair consisting of an $A_1$-module $F_1$ and an $A_2$-module $F_2$, let $E_1$ be the A-module $(p_1) * (F_1)$, $E_2$ the A-module $(p_2) * (F_2)$, $p_1$ and $p_2$ being the projections of A onto $A_1$ and $A_2$ respectively; then in the A-module $E = E_1 \oplus E_2,\ E_1 = e_1E,\ E_2 = e_2E$. The study of A-modules is thus reduced to that of $A_1$-modules and that of $A_2$-modules. In particular, every submodule M of E is of the form $M_1 \oplus M_2$, where $M_1 = e_1M$ and $M_2 = e_2M$.