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Math Help - prodct of principal ideal rings

  1. #1
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    prodct of principal ideal rings

    If A,B are principal ideal rings, then A \times B is also principal ideal ring.
    Is this right? Why??
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  2. #2
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    yes, if A and B both have 1 of course. the reason is that every ideal of A \times B is in the form I \times J for some ideal I of A and some ideal J of B.

    maybe i don't have to mention that A \times B is never a domain. so the result doesn't hold for principal ideal "domains".
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  3. #3
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    Umm...
    Let I=(a) \times (b) be an ideal in A \times B.
    Then how can I find a generator c s.t. I=(c)???
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  4. #4
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    well, assuming that your rings are commutative of course, if I = Aa and J = Bb, then I \times J = (A \times B)(a, b).
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