Generalization of quasi-Koszul algebras

In this paper, we generalize two kinds of graded algebras, δ-Koszul algebras and K _p algebras, to the non-graded cases. The trivial modules of δ-Koszul algebras have pure resolutions, while those of K _p algebras admit non-pure resolutions. We provide necessary and sufficient conditions for a notherian semiperfect algebra either to be a quasi-δ-Koszul algebra or to be a quasi-K _p algebra.