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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?

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I changed {\rm Min} to \min and {\rm dim} to \dim. The latter not only appears unitalicized, but also, unlike the former, automatically has proper spacing in things like $a\dim b$. Likewise the former, with the additional feature that in a "displayed", as opposed to "inline", setting, subscripts are positioned as in the following example: $\displaystyle\min_{x\in S}$. –  Michael Hardy May 31 '13 at 19:37
    
@ Michael Hardy: Thanks for editing but as much as I know $\min R$ denotes the minimal elements of $R$ as a set but ${\rm Min}R$ is the set of minimal prime ideals. –  QED Jun 1 '13 at 17:07
    
OK, I've changed it from {\rm Min} to \operatorname{Min}. –  Michael Hardy Jun 4 '13 at 20:45

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