We first characterize $tau$-complemented modules with relative (pre)-covers. We also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.

background: benzene is one of the most widely used volatile organic compounds showing severely adverse effects on human health. removing benzene from the airflow or controlling its amounts, is one of the issues put to discussion in the field of environmental engineering. the objective of this study was to examine the efficacy of the ultraviolet-c (uv-c) radiations on removing benzene from the a...

We investigate relative holomorphic connections on a principal bundle over family of compact complex manifolds. A sufficient condition is given for the existence connection analytic family. also introduce notion equivariant bundles and establish its relation with bundles.