Flexibility is a concept associated with variable needs of human beings.Today transformable structures are used to meet human changeable demands with different purposes and functions such as facades, roofs, sunshades, etc.A Flexible structure must be able to change physically and adapt to predetermined conditions.To provide flexibility and adaptability,transformable systems are required to poss...

In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question whether such can be extended $n$-Tor-rigid modules, and obtain an affirmative answer $2$-Tor-rigid that are generically free. Furthermore, in appendix, use Dao's eta function determine new classes hypersurfaces quotient un...

Let Uq denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that q is a complex root of unity of odd order and that Uq is obtained via Lusztig’s q-divided powers construction. We prove that all regular projective (tilting) modules for Uq are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl m...