We initiate the study of 1-torsion of finite modules over two-sided noetherian semiperfect rings. In particular, we give a criterion for determining when the 1-torsion submodule contains minimal generators of the module. We also provide an explicit construction for a projective cover of the submodule generated by the torsion elements in the top of the module. Some of the obtained results hold w...

We previously showed that the inverse limit of standard-graded polynomial rings with perfect (or semiperfect) coefficient field is a ring in an uncountable number variables. In this paper, we show result holds no hypothesis on field. also prove analogous for ultraproducts rings.