Exercise 1, § 5

Let M be an A-module and N a direct factor submodule of M.

a) Prove that N is stable under the bicommutant $A''_M$ of M.

b) Prove that every A-linear mapping from N to M is $A''_M$-linear.

c) Prove that restriction to N defines a ring homomorphism $A''_M\rightarrow A''_N$.

d) Suppose that $M/N$ is a sum of submodules isomorphic to quotients of N. Prove that the canonical homomorphism $A''_M\rightarrow A''_N$ is injective. If N is balanced, the same holds for M.