Exercise 13, § 5

Let $k$ be a ring and $n$ an integer $\geqslant 2$; denote by A the subring of $\mathbf{M}_n(k)$ consisting of the upper-triangular matrices and by M the right A-module $k^n$. Prove that M is projective and finitely generated but that the ring of homotheties of M is not dense in its bicommutant (cf. VIII, p. 88, Theorem 3 and VIII, p. 88, Remark). In particular, M is not balanced.