Prove that a field/commutative ring is IBN (invariant basis number).

I am searching all over and it keeps coming up how trivial it is, but I don't actually see the proof of a field being IBN. I would also like to know where a proof of a nonzero commutative ring is IBN can be found, because I keep finding in books that it is simply used as an example. I'm fine with receiving some bibliography.