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 .

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- Jun 18th 2013, 08:13 AMjohn27Let 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 . - Jun 18th 2013, 11:30 AMDrexel28Re: 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 - Jun 19th 2013, 05:50 AMjohn27Re: 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 AMHallsofIvyRe: 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.