Is Idempotent Equivalence in Rings Transitive?

[cogito²]

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.

 

[gel]

If P~Q and Q~R then, P=XY, Q=YX and Q=VW, R=WV.
Then,
P=P²=XYXY=XQY=(XV)(WY)
R=R²=WVWV=WQV=(WY)(XV)
so P~R.

 

[cogito²]

Alright that's about as complicated as I expected it to be...I basically had that written down, but apparently I don't quite have a fully functioning brain and for some reason couldn't see it. Many thanks.