Torsion intersection criterion for bound quiver algebras

Characterize the bound quiver algebras for which every nonzero nilpotent two-sided ideal $J$ satisfies $Tors_A(J)\cap Sub(A/J)\neq\{0\}$, and determine whether this property can be recognized from the path poset and minimal relations without enumerating two-sided ideals.

Background

The paper introduces a criterion asserting that if every nonzero nilpotent ideal has a nonzero module in $Tors_A(J)\cap Sub(A/J)$, then every self-orthogonal -tilting module is 1-tilting. This gives an algebra-level sufficient condition for resolving the self-orthogonal -tilting conjecture in a given class.

The open problem asks for a structural characterization of bound quiver algebras satisfying this criterion and, more specifically, for a recognition method based on the path poset and minimal relations rather than an exhaustive enumeration of two-sided ideals.