Exercise 2, § 5

Let K be a commutative field and A the subring of $\mathbf{M}_3(K)$ consisting of the matrices of the form

$a$ 0 0

$bc$ 0

0 0 $a$

for $a, b, c$ in K. Let M be the A-module $K^3$; it is the direct sum of the submodules $N = Ke_1+ Ke_2$ and $P = Ke_3$.

a) Prove that M is balanced, that the canonical mapping $A''_M\rightarrow A''_N$ (Exercise 1) is injective but not surjective, and that the canonical mapping $A''_M\rightarrow A''_P$ is surjective but not injective.

b) Deduce an example of a module Q over a ring B and a direct factor submodule R such that the canonical mapping $B''_Q\rightarrow B''_R$ is neither injective nor surjective (take $B = A\times A$ and $Q = M\times M$).