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Thread: Abstract-Ideals&Zorn's lemma

  1. #1
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    Abstract-Ideals&Zorn's lemma

    Hey there,
    I received this question:
    Let R be a commutative ring with identity 1 and let S be a sub-monoid of
    R-{0}.

    Use Zorn's lemma to prove that there is a maximal ideal J such as JnS=Empty Set.

    I proved a stronger proposition that says that in R, every proper ideal is contained in a maximal ideal. But is it equivalent to the proposition given in the question? How can I continue?

    TNX a lot!
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  2. #2
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    Quote Originally Posted by WannaBe View Post
    Hey there,
    I received this question:
    Let R be a commutative ring with identity 1 and let S be a sub-monoid of
    R-{0}.

    Use Zorn's lemma to prove that there is a maximal ideal J such as JnS=Empty Set.

    I proved a stronger proposition that says that in R, every proper ideal is contained in a maximal ideal. But is it equivalent to the proposition given in the question? How can I continue?

    TNX a lot!


    No, it is not equivalent but it's exactly the same idea and the proof is almost identical: define $\displaystyle D:=\{I\le R\;;\;I\cap S=\emptyset\}$.
    Now, $\displaystyle D\neq\emptyset$ (why??), and we can partial-order it by inclusion. Now prove that Zorn's lemma applies here and thus there's a maximal element in $\displaystyle D$ and voila...

    Tonio
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  3. #3
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    Hi

    With a very similar proof to the one of your new proposition, you can show that, given any sub-mono´d $\displaystyle S$ of $\displaystyle R-\{0\}$ and any ideal $\displaystyle I$ of $\displaystyle R$ such that $\displaystyle I\cap S=\emptyset,$ there is an ideal $\displaystyle J$ containing $\displaystyle I$ such that $\displaystyle J\cap S=\emptyset $ maximal for this property.

    This proposition is stronger than Krull's theorem, just apply it to $\displaystyle S=\{1\}$ and you proper ideal $\displaystyle I$, it will be easy to show that the ideal $\displaystyle J$ is a maximal ideal (which is not always true when $\displaystyle S\neq\{1\},$ what you have is that J is a prime ideal)
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  4. #4
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    Tnx a lot to you both, but how can we tell that every one of the ideals I such as InS=Empty SET is contained in another ideal with this proprety?
    I mean, let's take I1 AND I2 such as I1nS=phi and I2nS = phi... How can we tell that I1 is in I2 or I2 is in I1? Maybe they have no common elements?

    TNX!
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  5. #5
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    Quote Originally Posted by WannaBe View Post
    Tnx a lot to you both, but how can we tell that every one of the ideals I such as InS=Empty SET is contained in another ideal with this proprety?
    I mean, let's take I1 AND I2 such as I1nS=phi and I2nS = phi... How can we tell that I1 is in I2 or I2 is in I1? Maybe they have no common elements?

    TNX!

    True, maybe they don't...but you don't have to worry about this: Zorn's Lemma tells you to take a chain C in D (i.e., a totally ordered subset of D ), and then THERE you'll have that $\displaystyle I_1\,,\,I_2\in C\Longrightarrow I_1\subset I_2\,\,\,or\,\,\,I_2\subset I_1$

    Tonio
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  6. #6
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    Yep...TNX a lot to you both!
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  7. #7
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    I have one more question about this assignment:
    In the next part of the question I've proved that J is a prime ideal...]
    I now have to prove that if a is not a nilpotent element of R then there is a prime ideal I such as In{a^n|n is in N}=empty set...
    It's pretty obvius that I NEED to prove that the set {a^n|n is in N} is a monoid...All I need is to know why this set must contain the multiplication identity element ( 1)...

    HELP IS NEEDED

    TNX a lot!
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