On -u-S-flat modules and nonnil-u-S-injective modules

This paper introduces and studies the phi-u-S-flat (resp., nonnil-u-S-injective) modules, which are a generalization of both phi-flat modules and u-S-flat modules (resp., both nonnil-injective modules and u-S-injective modules). We give the Cartan-Eilenberg-Bass theorem for nonnil-u-S-Noetherian rings. Finally, we offer some new characterizations of the phi-von Neumann regular ring.