Effect of nilpotency on semisimplicity and cohomology of the $\mathbb{Z}$-module $\mathbb{Z}\mathrm{/}(p_1^{k_1}\times \cdots \times p_n^{k_n})\mathbb{Z}$

We show that existence of nonzero nilpotent elements in the $\mathbb{Z}$-module $\mathbb{Z}\mathrm{/}(p_1^{k_1}\times \cdots \times p_n^{k_n})\mathbb{Z}$ inhibits the module from possessing good structural properties. In particular, it stops it from being semisimple and from admitting certain good homological properties.