A weakly distance-regular digraph is $P$-polynomial if its attached scheme $P$-polynomial. In this paper, we characterize all digraphs.

For a left and right Noetherian ring $R$, we give some equivalent characterizations for $\_RR$ satisfying the Auslander condition in terms of flat (resp. injective) dimensions minimal injective coresolution resolution) $R$-modules. Furthermore, prove that an artin algebra $R$ condition, is Gorenstein if only subcategory consisting finitely generated modules contravariantly finite. As applicatio...

Several characterizations are given of (Zelmanowitz) regular modules among the torsionless modules, the locally projective modules, the nonsingular modules, and modules where certain submodules are pure. Along the way, a version of the unimodular row lemma for torsionless modules is given, and it is shown that a regular ring is left self-injective if and only if every nonsingular left module is...

For a ring R, call a class C of R-modules (pure-) mono-correct if for any M,N ∈ C the existence of (pure) monomorphisms M → N and N → M implies M ' N . Extending results and ideas of Rososhek from rings to modules, it is shown that, for an R-module M , the class σ[M ] of all M -subgenerated modules is mono-correct if and only if M is semisimple, and the class of all weakly M -injective modules ...