Is it true that given a local noetherian ring $(R, m)$ that modding out by a regular element drops the dimension by 1? I may be misunderstanding my professor, but he claimed that as long as you choose an element that avoids the minimal primes and $m^2$, this should work.
I do know that it will drop the dimension by at least one by Krull's principal ideal theorem, but I can't see why it is exactly one.