Exercise 14, § 5

What is the bicommutant of the $\mathbf{Z}$-module $\mathbf{Q}$? Prove that $\mathbf{Q}$ is a simple module over its bicommutant but that the ring of homotheties of the $\mathbf{Z}$-module $\mathbf{Q}$ is not dense in its bicommutant.