Exercise 10, § 3

Let $\rho : A\rightarrow B$ be a ring homomorphism used to view B as a right A-module. Suppose that the left B-module Hom$_A(B,A)$ is isomorphic to $B_s$.

a) Let M be a B-module, and let $\varphi : B_s\rightarrow$ Hom$_A(B,A)$ be an isomorphism. Prove that the mapping $u\mapsto \varphi (1)\circ u$ from Hom$_B(M,B)$ to Hom$_A(M,A)$ is bijective.

b) Suppose, moreover, that the A-module B is finitely generated. The ring B is autoinjective (Exercise 8) if and only if the same holds for A.

c) Let G be a finite group. Show that the conditions above are satisfied when B is taken to be the A-module A[G] endowed with the ring structure such that $(ae_g)(a'e_{g'}) =aa'e_{gg'}$ for $a, a'$ in A and $g, g'$ in G. In particular, if K is a commutative field, then the algebra K[G] is autoinjective.