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

• Jun 18th 2013, 08:13 AM
john27
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
$A_{mxn}B_{nxm}=I_m$ and $B_{nxm}A_{mxn}=I_n$ then m=n.

Im and In are identity marix.

Thank you .
• Jun 18th 2013, 11:30 AM
Drexel28
Re: Let A be a ring, prove that if AB=I and BA=I .....
Think about it like this. You have shown that $A$ is an isomorphism of $R^n\to R^m$. Assuming that $R$ is commutative (so that it has the invariant basis number property) then $n=m$.

Best,
Alex
• Jun 19th 2013, 05:50 AM
john27
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 ?
• Jun 19th 2013, 07:10 AM
HallsofIvy
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.