On a Generalization of Weakly Supplemented Modules

In this paper, over an arbitrary ring we define the notion of weakly radical supplemented modules (or briefly wrs-module), which is adapted from Zöschinger’s radical supplemented modules over a discrete valuation ring (DVR), and obtain the various properties of these modules. We prove that a wrs-module having a small radical is weakly supplemented. Moreover, we show that a ring R is left perfect if and only if every left R-module is wrs. Also, we prove that every wrs-module over a DVR is radical supplemented.