T-quasi-continuous modules

A module M is called t-quasi-continuous if M is t-extending and whenever A and B are nonsingular direct summands of M with , then is a direct summand of M. The class of t-quasi-continuous modules lies properly between the classes of t-continuous modules and t-extending modules. Moreover, the concept of a t-quasi-continuous module generalizes quasi-continuous modules. In this article, we give a comprehensive study of t-quasi-continuous modules.