A note regarding some generalization of a Schur's theorem

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.