On an upper bound for the global dimension of Auslander--Dlab--Ringel algebras

Lin and Xi introduced Auslander–Dlab–Ringel (ADR) algebras of seimlocal modules as a generalization of original ADR algebras and showed that they are quasi-hereditary. In this paper, we prove that such algebras are always left-strongly quasi-hereditary. As an application, we give a better upper bound for global dimension of ADR algebras of semilocal modules. Moreover we describe characterizations of original ADR algebras to be strongly quasi-hereditary.