LAUSR

In this manuscript, we define the class of ${\omega }_1$ -weakly $\alpha$ -projective QTAG-modules for the infinite ordinal $\alpha$ and provide its systematic study for the finite ordinal. Furthermore, we generalize this class to $\left(\omega .2+n\right)$ -projective modules and obtain some characterizations. We also study the $\omega$ -totally weak $\left(\omega .2+n\right)$ -projective modules under the formation of ${\omega }_1$ -bijections.