A note regarding some generalization of a Schur's theorem

TL;DR

This paper provides a concise proof of a generalization related to Schur's theorem, simplifying a previously lengthy proof in the context of infinite groups.

Contribution

It introduces a simplified and more accessible proof of a generalization of Schur's theorem for infinite groups.

Findings

Simplified proof of the main theorem

Enhanced understanding of the upper central series

Clarification of the generalization of Schur's theorem

Abstract

The article on the upper central series of infinite groups by M. de Falco, F. de Giovanni, C. Musella and Y.P. Sysak, proceedings of the american mathematical society, Volume 139, Number 2, February 2011, 385--389 consists of a quite long proof of a main theorem. We offer a simple and very brief proof of this result.