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

Abstract

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 from possessing good structural properties. In particular, it stops it from being semisimple and from admitting certain good homological properties.