  Ringel's right-strongly quasi-hereditary algebras are a distinguished class
of quasi-hereditary algebras of Cline-Parshall-Scott. We give characterizations
of these algebras in terms of heredity chains and right rejective
subcategories. We prove that any artin algebra of global dimension at most two
is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra
of a representation-finite algebra $A$ is strongly quasi-hereditary if and only
if $A$ is a Nakayama algebra.
