Title:  Classes of modules relative to torsion theories

Abstract:  Our purpose is extend known results of some classes of modules to torsion theoretic setting in a way so that the former results are recovered when some torsion theory is chosen. For example we generalize the concept of relative pure injectivity to relative pure $\tau$-injectivity, where $\tau$ is a given hereditary torsion theory. If $\tau$ is the improper torsion theory then relative pure injectivity and relative pure $\tau$-injectivity are equivalent. This new concept retains some of the important properties of pure injectives. For instance, we show that the class of pure $\tau$-injective modules is enveloping.