I'm curious if there exists a class of rings generalizing quasi-unmixed rings. I guess a generalization of quasi-unmixed rings can be done as follows: For a fixed integer $i$ $$\forall p\in\operatorname{Min} R,\quad \dim R- \dim R/p\leq i$$ ($i=0$ gives the quasi-unmixed rings.)

**Have such rings been studied and characterized?**