Exercise 1, § 12

Let A and B be K-algebras and M an A-module; view $M\otimes B$ as a module over the algebra $C = A\otimes B$. Identify M with an A-submodule of $M\otimes B$.

a) Prove the inclusion $M\cap \mathfrak{R}_C(M\otimes B)\subset \mathfrak{R}_A(M)$ (consider an A-linear homomorphism from M to a simple A-module).

b) Prove that we have equality in each of the following cases:

(i) A is finite-dimensional over K.

(ii) The A-module M is finitely generated, and A is the union of a right directed family of subalgebras that are finite-dimensional over K.

(iii) $M = A_s$, and the radical of A is nilpotent.

(In case (i), one proves that every homomorphism from $M\otimes B$ to a simple C-module S is zero on $\mathfrak{R}_A$(M), by observing that $\mathfrak{R}_C(S)$ is zero.)