One of my books defines a relation which is "evidently" an equivalence relation. It says that two idempotents in a ring P and Q are said to be equivalent if there exist elements X and Y such that P = XY and Q = YX. The proof that this relation is transitive eludes me. There is so little information, that I feel like this has to have a really short proof, but I just can't seem to figure it out (or find it on the magical internet). If anyone can can ease my frustration, I would be grateful.