Rulings for quotients by normal quadratic elements

Develop a framework for rulings on the noncommutative projective quadric Proj A where A = S/w and w ∈ S_2 is a normal quadratic regular element (not necessarily central), and determine how such rulings relate to line modules over A and over S.

Background

Smith–Van den Bergh treat rulings for central quadratic hypersurfaces and often assume domain conditions on the quotient. The algebra S exhibits non-central normal quadratic elements, raising the need to generalize ruling notions to such quotients.

This question seeks both conceptual development and concrete relationships to line modules, aiming to unify geometric and homological structures in the broader normal (noncentral) setting.