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Math Help - Question about ideals and integral domains.

  1. #1
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    Question about ideals and integral domains.

    Suppose that A is a ring and I is an ideal of A. Prove that the quotient ring A/I is an integral domain if and only if I satisfies the following:
    I \neq A and xy \in I \implies (x\in I or  y\in I).

    I have tried it both ways using a contradictory argument but to no avail. Help much appreciated.
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  2. #2
    Super Member girdav's Avatar
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    Re: Question about ideals and integral domains.

    An ideal which satisfies the second condition is said to be prime. Use the fact that x\in I is the same thing as the class of x modulo I is the class of 0, and the product of two classes is the class of the product.
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  3. #3
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    Re: Question about ideals and integral domains.

    this is pretty basic: suppose I is a prime ideal of A. then if (x+I)(y+I) = I in A/I, and x is not in I, then xy + I = I, so xy is in I, and since I is prime, and x is not in I, y is in I.

    but this means that y+I = I, that is, A/I has no zero divisors. provided that A was a commutative ring with unity in the first place, A/I is an integral domain (some authors do not require commutativity nor an unit).

    (the condition I ≠ A ensures we have some other element besides I = 0 + I in A/I, so that we have non-zero divisors at all).

    the converse is proven similarly: using a direct approach, rather than contradiction, works well.
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