Exercise 6, § 6

Let A and B be rings. Suppose given

– a B-module F(M) for every A-module M and

– a B-linear homomorphism $F(u) : F(M)\rightarrow F(N)$ for every homomorphism of A-modules $u: M\rightarrow N$,

such that following properties hold:

(i) We have $F(1_M) = 1_{F(M)}$ for every A-module M.

(ii) For $u\in$ Hom$_A(M,N)$ and $v\in$ Hom$_A(N,P)$, we have $F(v\circ u) = F(v)\circ F(u)$.

(iii) For $u, v\in$ Hom$_A(M,N)$, we have $F(u+v) = F(u) + F(v)$.

a) Prove that for every A-module M, the mapping $u\mapsto F(u)$ from End$_A(M)$ to End$_B(F(M))$ is a ring homomorphism. Deduce a $(B$, A)-bimodule structure on $F(A_s)$ from this.

b) Let M be an A-module; for $m\in M$, denote by $h_m: A_s\rightarrow M$ the homomorphism $a\mapsto am$. Prove that there exists a unique B-linear homomorphism $\theta_M$ from $F(A_s)\otimes_AM$ to F(M) such that $\theta_M(x\otimes m) = F(h_m)(x)$ for every $x\in F(A_s)$ and $m\in M$. We have $F(u)\circ \theta_M=\theta_N\circ (1_{F(A_s)}\otimes u)$ for every homomorphism of A-modules $u: M\rightarrow N$.