Strongly $\phi$-flat modules, strongly nonnil-injective modules and their homological dimensions

In this paper, we first introduce and study the notions of strongly $\phi$-flat modules and strongly nonnil-injective modules. And then, we investigate the homology dimensions of modules and rings in terms of these two notions. Finally we give a new homological characterizations of $\phi$-Dedekind rings and $\phi$-\Prufer\ rings.