Definition 2

Let E be a (unital) A-algebra. A left (resp. right) E-module is a left (resp. right) module over the underlying ring of E.

Unless otherwise mentioned, all the modules and multimodules considered in this no. are left modules and multimodules.

If M is an E-module, the homomorphism $\eta : A \to E$ (§ 1, no. 4) then defines on M an A-module structure, said to be underlying the E-module structure on M; for $\alpha \in A, s \in E, x \in M$,

so that for all $s \in E$, the homothety $h_s : x \mapsto sx$ of M is an endomorphism of the underlying A-module structure. Conversely, being given an E-module structure on M is equivalent to being given an A-module structure on M and an A-algebra homomorphism $s \mapsto h_s$ of E into $\mathrm{End}_A(M)$.