2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106441With a grading previously introduced by the second-named author, the multiplication maps in the preprojective algebra satisfy a maximal rank property that is similar to the maximal rank property proven by Hochster and Laksov for the multiplication maps in the commutative polynomial ring. The result follows from a more general theorem about the maximal rank property of a minimal almost split morphism, which also yields a quadratic inequality for the dimensions of indecomposable modules involved.Representation TheoryCommutative AlgebraAlgebraic Geometry13D02; 16G30; 16G70 (Primary)text