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Thread: Union ideals

  1. #1
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    Union ideals

    Hl!. I have problems with exercice


    Let $\displaystyle I$ and $\displaystyle J$ be ideals in a commutative ring $\displaystyle R$ . Prove that their union $\displaystyle I \cup{J}$ is a ideal if and only if $\displaystyle I\subseteq{J}$ or $\displaystyle J\subseteq{I}$

    Hint please


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  2. #2
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    Re: Union ideals

    If y is in J but not in I and x is in I but not in J, consider x+y
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  3. #3
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    Re: Union ideals

    Suppose i is in I but not in J. Suppose j is in J but not in I.

    If $\displaystyle I\cup J$ is an ideal, then $\displaystyle i+j\in I\cup J$, but $\displaystyle i+j\in I$ implies ________ and $\displaystyle i+j\in J$ implies _________. Any contradictions?

    I will admit my knowledge of commutative algebra is less than ideal these days.
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  4. #4
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    Re: Union ideals

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