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

TL;DR

This paper investigates how nilpotent elements in specific $ ext{Z}$-modules affect their structural and homological properties, revealing that nilpotency prevents semisimplicity and desirable cohomological features.

Contribution

It demonstrates the impact of nilpotent elements on the semisimplicity and cohomological properties of certain $ ext{Z}$-modules, providing new insights into their algebraic structure.

Findings

Nilpotent elements inhibit semisimplicity.

Nilpotency prevents certain homological properties.

Modules with nilpotent elements lack good structural features.

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.