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Thread: prodct of principal ideal rings

  1. #1
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    prodct of principal ideal rings

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

    maybe i don't have to mention that $\displaystyle A \times B$ 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 $\displaystyle I=(a) \times (b)$ be an ideal in $\displaystyle A \times B$.
    Then how can I find a generator $\displaystyle c$ s.t. $\displaystyle I=(c)$???
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  4. #4
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    well, assuming that your rings are commutative of course, if $\displaystyle I = Aa$ and $\displaystyle J = Bb$, then $\displaystyle I \times J = (A \times B)(a, b).$
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