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Math Help - Let A be a ring, prove that if AB=I and BA=I .....

  1. #1
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    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 .
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  2. #2
    MHF Contributor Drexel28's Avatar
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    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
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  3. #3
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    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 ?
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  4. #4
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    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.
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