Re: Let A be a ring, prove that if AB=I and BA=I .....
Think about it like this. You have shown that is an isomorphism of . Assuming that is commutative (so that it has the invariant basis number property) then .
Re: Let A be a ring, prove that if AB=I and BA=I .....
Yes, that is what Drexel28 said. By the way, you should never use the same symbol to mean two different things- here you use "A" to mean both the ring and an element of the ring.