I am considering the following problem：
Let $ A $ be a commutative Noetherian local ring, $ I $, $ I' $ be two ideals of $ A $ satisfying that $ \dim(A/I) = \dim(A/I') $. Is it true that $ \dim(A/I)=\dim(A/I\cap I') $ always holds? (Here $ \dim $ means Krull dimension.)
Any ideas would be thankful.
Yes, with no Noetherian or local hypothesis. This is because if a prime ideal contains $I \cap J$ it contains $IJ$, and hence contains either $I$ or $J$ (so any chain containing $I \cap J$ must contain either $I$ or $J$). To see this, suppose $P$ is prime and $IJ \subseteq P$ but both $I$ and $J$ are not contained in $P$. Then there exist $x\in I$, $y\in J$ such that $x\notin P$ and $y\notin P$. But as $xy\in IJ$ we have $xy \in P$, contradicting primality of $P$