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Math Help - Rings with 0 = 1.

  1. #1
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    Rings with 0 = 1.

    Hi:
    Let R be a ring with unity and assume 0,1 are the identity elements for addition and multiplication respectively in  R. Can 0=1 be true? I think the only way is R=(0). Any hint will be welcome. Regards.
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  2. #2
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    Quote Originally Posted by ENRIQUESTEFANINI View Post
    Hi:
    Let R be a ring with unity and assume 0,1 are the identity elements for addition and multiplication respectively in  R. Can 0=1 be true? I think the only way is R=(0). Any hint will be welcome. Regards.

    Indeed, and that is why many authors disqualify 0 = 1 or explicitly require that these are two different elements when dealing with rings.

    Tonio
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  3. #3
    Senior Member Shanks's Avatar
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    Yes, If 0=1 in R, then R is trivially {0}, containning only the element 0.
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by ENRIQUESTEFANINI View Post
    Hi:
    Let R be a ring with unity and assume 0,1 are the identity elements for addition and multiplication respectively in  R. Can 0=1 be true? I think the only way is R=(0). Any hint will be welcome. Regards.
    The hint to proving it is a.1= \ldots
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  5. #5
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    Thanks, thanks. I'd seen definitions requiring the ring to have two or more elements, but I'd never seen the same thing specified as zero and one being diferent elements. This seemed a bit odd to me. Now I see that authors use these two constrains interchangably. Thanks again,

    Enrique.
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