On weak and $\omega$-high purity in the category of modules

In the category of right unitary modules over the associative ring $R$ with $1$, one can define weak $\frak F$ purity, where $\frak F$ is the set of right ideals of $R$ satisfying certain conditions. This is a generalization of the concept of neatness in Abelian group theory. Using the properties of weak $\frak F$-purity, several classes of rings can be characterized. Moreover, an affirmative answer can be given to question 18 [question 14 in the English translation] of A. P. Mishina and L. A. Skornyakov's book “Abelian groups and modules”, which deals with properties of $\omega$-high purity. Groups of weakly $\frak F$-pure and $\omega$-high extensions are studied.
Bibliography: 15 titles.