Height unmixed ideal and a non-zero divisor 
Let $R$ be a commutative Noetherian ring with unit and  $I$ an unmixed ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an unmixed ideal?

Background:
A proper ideal $I$ in a Noetherian ring $R$ is said to be unmixed 
if the heights of its prime divisors are all equal, i.e., $\operatorname{height} I=\operatorname{height}\mathfrak{p}$ for all $\mathfrak{p}\in \operatorname{Ass} I$.

Let $R$ be a noetherian integral domain and $I=(0)$. If $\dim R=2$ and $R$ is not Cohen-Macaulay, then there is $x\in R$, $x\ne 0$, such that $xR$ is not unmixed. (For more details look here.)
