In our class today, we were arguing about getting too much easy work. So now we had to solve this impossible question that not even the teacher could solve. THIS IS CRAZY!
If M ⊗R N ≠ 0 has finite length, then the Krull dimension of N (i.e., the dimension of R modulo the annihilator of N) is at most the projective dimension of M.
Edit: Apparently, THIS PROBLEM HAS NEVER EVER BEEN SOLVED.