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Math Help - Non-Zero Rings

  1. #1
    pkr
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    Non-Zero Rings



    This question confuses me in many ways, by "non-zero" is it meant a ring with no divisors of zero?

    As an ID is a commutative ring-with-a-1 no divisors of zero, I guess i'm going to have to show there does exist a divisor of zero in R1 X R2?

    Not sure at all how to go about starting this one...
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  2. #2
    Senior Member JaneBennet's Avatar
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    It depends on how you define rings. Some authors insist that a ring must contain a nonzero multiplicative identity. Others do not insist on this condition. If you use the second alternative, then it is possible for a ring to contain just one element, namely the zero element.

    A nonzero ring would then be a ring that contains at least one nonzero element.

    Here’s a hint to get you started. Let a_1 be a nonzero element in R_1 and a_2 be a nonzero element in R_2. Then (a_1,0) and (0,a_2) are nonzero elements in R_1\times R_2. Now multiply them together. What do you get?
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  3. #3
    pkr
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    You would get (0,0) in R1 X R2 which would imply this ring has a element, say Q, such that Q multiplied by any element = Q, so it isn't an integral domain?

    Obviously i'd write out the proof more concise than, but am I getting at the right idea?
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  4. #4
    Senior Member JaneBennet's Avatar
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    Quote Originally Posted by pkr View Post
    You would get (0,0) in R1 X R2 which would imply this ring has a element, say Q, such that Q multiplied by any element = Q, so it isn't an integral domain?
    What is the definition of an integral domain?
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  5. #5
    pkr
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    Commutative ring with a 1, no divisors of zero.

    So, because a1 and a2, are non-zero elements of R1 and R2, and R1 X R2 = (0, 0) implies R1 X R2 does posses a divisor of zero, and thus cannot be a integral domain.
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