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Math Help - Prime Ideal and Integral Domain

  1. #1
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    Prime Ideal and Integral Domain

    I was given the following question. Let A be a commutative ring with unity, and J an ideal of A. Prove J is a prime ideal if and only if A/J is an integral domain.

    This is how I proved it but I dont know if it is complete enough. If anyone has any suggestions please let me know

    Suppose that there exists a,b $in$ A/J, so that ab = 0. This also means that ab $in$ J. We need to prove that there are no zero divisors in A/J. However, since J is a prime ideal, it means that a $in$ J or b $in$ J. Since a $in$ J or b $in$ J, this means that a = 0 or b = 0. Thus there means that there are no zero divisors. Hence A/J is an integral domain.

    Suppose A/J is an integral domain. Let ab $in$ A. Since A/J is an integral domain it has no zero divisors, this means that ab = 0 and either a = 0 or b = 0. Which means that a $in$ J or b $in$ J and ab $in$ J. Thus J is a prime ideal.
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  2. #2
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    Re: Prime Ideal and Integral Domain

    you're confusing a and b with the COSETS a+J and b+J. the idea of your proof is correct, but you should be more careful.

    suppose J is a prime ideal, and that (a+J)(b+J) = 0+J = J in A/J. this means that ab+J = J, hence ab is in J. Since J is a prime ideal, either a is in J, or b is in J. if a is in J, then a+J = J = 0+J, and similarly for b, so that A/J is an integral domain.

    similarly, if we have ab in J, so that ab+J = 0 J = J, then either a+J = 0+J, so that a is in J, or b+J = 0+J, so that b is in J, hence J is a prime ideal.
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  3. #3
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    Re: Prime Ideal and Integral Domain

    Find the number of ideals in a ring Zn where n = product of prime numbers???
    I want the answer any body help me
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  4. #4
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    Re: Prime Ideal and Integral Domain

    start by looking at the subgroups of the additive group of Zn. are any of these ideals?
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