Exercise 1, § 11

Let $p$ be a prime number. Denote by $\mathscr{C}$ the set of classes of $\mathbf{Z}$-modules of the form $(\mathbf{Z}/p^2\mathbf{Z})^n\oplus (\mathbf{Z}/p^3\mathbf{Z})^m$ for $n, m\in \mathbf{N}$.

a) Prove that $\mathscr{C}$ is additive but not hereditary. What is the smallest hereditary set $\mathscr{H}$ containing $\mathscr{C}$?

b) Prove that every short exact sequence of modules of type $\mathscr{C}$ splits. Describe the group $K(\mathscr{C})$.

c) Construct an exact sequence

of modules of type $\mathscr{C}$ such that the classes of $M_0\oplus M_2\oplus M_4$ and $M_1\oplus M_3$ in $K(\mathscr{C})$ are distinct.

d) Is the natural homomorphism from $K(\mathscr{C})$ to $K(\mathscr{H})$ injective?