The General PBW Property

For ungraded quotients of an arbitrary $\mathbb{Z}$-graded ring, we define the general PBW property, that covers the classical PBW property and the $N$-type PBW property studied via the $N$-Koszulity by several authors ([BG1], BG2], [FV]). In view of the noncommutative Gr\"obner basis theory, we conclude that every ungraded quotient of a path algebra (or a free algebra) has the general PBW property. We remark that an earlier result of Golod [Gol] concerning Gr\"obner bases can be used to give a homological characterization of the general PBW property in terms of Shafarevich complex. Examples of application are given.