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Math Help - Ideals - Intersection of Ideals

  1. #1
    Super Member Bernhard's Avatar
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    Ideals - Intersection of Ideals

    If I and J are ideals of R, prove that their intersection  I \cap J is also an ideal of R

    Hence show that the intersection of an arbitrary nonempty is again an ideal (do not assume that the collection is countable)


    Would appreciate help with this exercise ... my main problem is with generalising to an uncountable collection

    Peter
    Last edited by Bernhard; March 30th 2013 at 06:31 PM.
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  2. #2
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    Re: Ideals - Intersection of Ideals

    Quote Originally Posted by Bernhard View Post
    If I and J are ideals of R, prove that their intersection  I \cap J is also an ideal of R

    Hence show that the intersection of an arbitrary nonempty is again an ideal (do not assume that the collection is countable)


    Would appreciate help with this exercise ... my main problem is with generalising to an uncountable collection

    Peter
    There should be nothing different in the uncountable case save for notation (which is possible to hide in a proof likes this). How did you prove the other cases?
    Thanks from Bernhard
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Ideals - Intersection of Ideals

    I and J ideals of R  \longrightarrow  I \cap J is an ideal of R

    Proof

     x, y \in I \cap J

     \longrightarrow  x, y \in I , x,y \in J

     \longrightarrow  x - y \in I , x - y \in J

     \longrightarrow  x - y \in I \cap J .... (1)



     x \in I \cap J

     \longrightarrow  x \in I , x \in J

     \longrightarrow  rx \in I , rx \in J for all  r \in R

     \longrightarrow  rx \in I \cap J ..... (1)

    (1) , (2)  \longrightarrow I \cap J is ideal of R



    Then since  I \cap J is an ideal, use the above proof on K and  I \cap J to show  I \cap J \cap K is an ideal and so on for countably many ideals.
    Last edited by Bernhard; March 31st 2013 at 05:06 PM.
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    Re: Ideals - Intersection of Ideals

    Given an uncountable indexing \Lambda of ideals, I modified your proof for the uncountable case. If you take \Lambda to be a countable or finite indexing, you only really need this one proof.

    Quote Originally Posted by Bernhard View Post
    Proof

     x, y \in \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha}

     \longrightarrow  x, y \in I_\alpha for all \alpha \in \Lambda

     \longrightarrow  x - y \in I_\alpha for all \alpha \in \Lambda

     \longrightarrow  x - y \in \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha} .... (1)



     x \in I \cap J

     \longrightarrow  x \in I_\alpha for all \alpha \in \Lambda

     \longrightarrow  rx \in I_\alpha for all \alpha \in \Lambda and for all  r \in R

     \longrightarrow  rx \in \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha} ..... (1)

    (1) , (2)  \longrightarrow  \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha} is ideal of R
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