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Math Help - M+A = A+A implies M = A?

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    M+A = A+A implies M = A?

    Let A be a commutative unital ring and M an A-module. Suppose that M\oplus A \cong A\oplus A. Is it true that necessary M\cong A?
    Last edited by Curu; January 16th 2013 at 01:11 PM.
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    Re: M+A = A+A implies M = A?

    The answer is affirmative and contained in these beautiful notes (click) by Brian Conrad. A proof an be find in Weibel's K-Book (click).
    Last edited by Curu; January 16th 2013 at 12:05 PM.
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    Re: M+A = A+A implies M = A?

    It doesn't really require a very complicated proof. In any ring, not just "commutative" or "unary" (in fact, since there is no mention of multiplication, you don't even need a ring), we have additive inverses. Adding the additive inverse of A to both sides immediately results in M= A.
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    Re: M+A = A+A implies M = A?

    Quote Originally Posted by HallsofIvy View Post
    It doesn't really require a very complicated proof. In any ring, not just "commutative" or "unary" (in fact, since there is no mention of multiplication, you don't even need a ring), we have additive inverses. Adding the additive inverse of A to both sides immediately results in M= A.
    I really think you misunderstood the question in the OP. Maybe I had to specify that with -\oplus - I mean the (bi)product (in the abelian category) of A-module, where A is supposed to be commutative and unital in order to easily satisfy the IBP.
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