Proposition 1

Let E, F, G be three A-modules, u a linear mapping of E into F and v a linear mapping of F into G. Then the composite mapping $v \circ u$ is linear.

(1) If u:E → F and v:F → E are two linear mappings such that v ∘ u is the identity mapping on E and u ∘ v is the identity mapping on F, u is an isomorphism of E onto F and v is the inverse isomorphism.

(2) Every bijective linear mapping u:E → F is an isomorphism of E onto F.

These propositions follow immediately from Definition 4.

Propositions 1 and 2 show that linear mappings can be taken as the morphisms for the species of A-module structures (Set Theory, IV, § 2, no. 1); we shall henceforth always assume that this choice of morphisms has been made.

Given two left (resp. right) A-modules E and F, let Hom(E, F) or Hom_A(E, F) denote the set of linear mappings of E into F.

The set Hom(E, F) is a commutative ring, a subgroup of the product commutative group F^E of all mappings of E into F (I, § 4, no. 8); recall that for two elements u, v of F^E and for all x ∈ E,

whence it follows immediately that, if u and v are linear, so are u + v and −u. If G is a third left (resp. right) A-module, f, f_1, f_2 elements of Hom(E, F) and g, g_1, g_2 elements of Hom(F, G), the following relations are immediately verified:

(6)

(7) (8)

In particular, the law of composition (f, g) ↦ f ∘ g on Hom(E, E) defines with the above additive group structure a ring structure on Hom(E, E) whose unit element, denoted by 1_E or Id_E, is the identity mapping on E; the linear mappings of E into itself are also called endomorphisms of the A-module E and the ring Hom(E, E) is also denoted by End(E) or End_A(E). The automorphisms of the A-module E are just the invertible elements of End(E) (Proposition 2); they form a multiplicative group, denoted by Aut(E) or GL(E) which is also called the linear group relative to E.

It follows from (6) and (7) that, for two A-modules E, F, Hom(E, F) has the canonical structure of a left module over the ring Hom(F, F) and of a right module over Hom(E, E).

Let E, F, E', F' be four (left) A-modules, u:E' → E and v:F → F' A-linear mappings. If every element f ∈ Hom(E, F) is associated with the element v ∘ f ∘ u ∈ Hom(E', F'), a mapping

is defined, which is $\mathbf{Z}$-linear and which is denoted by $\mathrm{Hom}(u, v)$ or $\mathrm{Hom}_A(u, v)$. If $u, u_1, u_2$ belong to $\mathrm{Hom}(E', E)$ and $v, v_1, v_2$ to $\mathrm{Hom}(F, F')$, then

Let $E'', F''$ be two $A$-modules, $u': E'' \to E'$ and $v': F' \to F''$ linear mappings. Then

If $u$ is an isomorphism of $E'$ onto $E$ and $v$ an isomorphism of $F$ onto $F'$, $\mathrm{Hom}(u, v)$ is an isomorphism of $\mathrm{Hom}(E, F)$ onto $\mathrm{Hom}(E', F')$ whose inverse isomorphism is $\mathrm{Hom}(u^{-1}, v^{-1})$ by (10).

If $h$ (resp. $k$) is an endomorphism of $E$ (resp. $F$), $\mathrm{Hom}(h, 1_F)$ (resp. $\mathrm{Hom}(1_E, k)$) is just the homothety with ratio $h$ (resp. $k$) for the right (resp. left) module structure on the ring $\mathrm{End}(E)$ (resp. $\mathrm{End}(F)$) defined above.