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

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