The left and the right parts of a module category

In this paper, we study, for an artin algebra, the class LA (and RA) which is a full subcategory of the category modA of finitely generated A-modules, and which consists of all indecomposable A-modules whose predecessors (and successors) have projective dimension (and injective dimension, respectively) at most one. We consider quotient algebras of A, which contain the information on these classes, then define and characterize those algebras for which the class is LA is contravariantly finite (and RA is covariantly finite, respectively).