# Thread: Let A be a ring, prove that if AB=I and BA=I .....

1. ## Let A be a ring, prove that if AB=I and BA=I .....

Can you help me prove this statements.

Let A be a ring, and A, B be 2 matrixs. if
$\displaystyle$A_{mxn}B_{nxm}=I_m$$and \displaystyle B_{nxm}A_{mxn}=I_n$$ then m=n.

Im and In are identity marix.

Thank you .

2. ## Re: Let A be a ring, prove that if AB=I and BA=I .....

Think about it like this. You have shown that $\displaystyle A$ is an isomorphism of $\displaystyle R^n\to R^m$. Assuming that $\displaystyle R$ is commutative (so that it has the invariant basis number property) then $\displaystyle n=m$.

Best,
Alex

3. ## Re: Let A be a ring, prove that if AB=I and BA=I .....

Yes A is a commutative Ring with identity . Is n=m ?

4. ## 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.