Injective modules, projective modules and their relationships to some rings

Abstract

The main objective of this study is to introduce several facts about the relationship of injective and projective modules with two rings namely H-ring and CO-H-ring. We extend some important results given H-ring and CO-H-ring. As applications we obtain some new results when M is a multiplication and faithful R-module, then R is H-ring. Also we investigate that any divisible module over Euclidean domain R gives R is H-ring. It is important to point out the fact that previous ring can be obtained from some modules over another ring namely QF-ring. Finally, we show that if M is a free module over the ring R, then R is CO-H-ring. To prove these results we have had to develop new methods.