Some time ago I found a theorem here at math.SE that states (under assumptions I can't recall):
Let M and N be two R-Modules, if $l(M\otimes L)=l(N\otimes L)$ for all R-modules $L$ then $M\simeq N$
Here $l(A)$ stands for the length of the R-module A. I have searched here alot but couldn't find any result that combines length and isomorphism. What am I asking for is for a reference of an article or book that provides me the conditions to make the above statement true.
Any help is appreciated