Fixed modules over skew group rings and their applications to the mod-retractability

In this paper, we investigate the fixed submodules of modules over skew group rings. We give two applications, the first concerns the determination of a simple basis of the center of a skew group ring. In the second application, we characterize in terms of the structure groups all the mod-retractable skew group rings. More precisely, we prove that if [Formula: see text] is a group acting on a field [Formula: see text] with kernel [Formula: see text], then the skew group ring [Formula: see text] is mod-retractable if and only if [Formula: see text] is mod-retractable and [Formula: see text] is of finite index in [Formula: see text].