Exercise 26, § 1

Let E be a free A-module with a basis containing at least two elements.

(a) Show that every semi-linear mapping of E into itself (relative to an automorphism of A) which permutes with all the automorphisms of E, is necessarily a homothety $x \mapsto \alpha x \ (\alpha \in A)$.

(b) Deduce from (a) that the centre of $\operatorname{End}_A(E)$ is the ring of central homotheties, isomorphic to the centre of A.

(c) Deduce from (a) that the centre of the group $\mathbf{GL}(E)$ is the group of invertible central homotheties of E, which is isomorphic to the multiplicative group of invertible elements of the centre of A.