Chapter VIII

Semisimple Modules and Rings

In this chapter, when we speak of a module (without further specification), we mean a left module. Let A be a ring and M an A-module. For any $a\in A$, we denote the homothety $x\mapsto ax$ of M by $a_M$. The mapping $a\mapsto a_M$ is a homomorphism from the ring A to a subring $A_M$ of End$_{\mathbf{Z}}(M)$, which we call the ring of homotheties of M.

Unless stated otherwise, the algebras we consider are associative and unital; by a subalgebra of an algebra E, we mean a subalgebra containing the unit element of E; the algebra homomorphisms are assumed to be unital.

Let K be a commutative ring and L a commutative K-algebra. For any K-module E, we denote the L-module $L\otimes_KE$ derived from it by extension of scalars by $E_{(L)}$ (II, §5, No. 1, p. 277). If A is a K-algebra and M a left A-module, then $A_{(L)}$ is endowed with a natural L-algebra structure (III, §1, No. 5, p. 433) and $M_{(L)}$ is endowed with the left $A_{(L)}$-module structure with law of action given by the formula $(\lambda \otimes a)(\mu\otimes m) =\lambda \mu\otimes am$.

For any commutative ring K and any group G, we denote the algebra $K^{(G)}$ of G over the ring K by K[G] (III, §2, No. 6, p. 446).