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Thread: Find ideals

  1. #1
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    Find ideals

    Hi!

    I have problem with this exercice

    Find three ideals (a) in  \in{\mathbb{Z}} with thr property that (24)\subsetneq (a) ( \subsetneq means is a proper subsets of)

    - Definition


    An ideal in a commutative ring R is a subset I of R such that




    a) 0\in{I}
    b) a,b \in{I} \Rightarrow{a+b \in{I}}
    c) If a\in{I}, r\in{A} then ra \in{I}

    I think of (2) (4) and (6) but are not ideal since (b)fails
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  2. #2
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    Re: Find ideals

    First, you mean " r\in R" rather than " r\in A", don't you? (2), (4), and (6) are the sets of all multiples of 2, 4, and 6, respectively, right? Then it certainly is true that "if a and b in I then a+ b in I". if a and b are multiples of r, then a= mr, and b= nr so a+ b= mr+ nr= (m+ n)r.
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  3. #3
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    Re: Find ideals

    Means r \in R , Second for example (4)=\{mr:\,r\in\mathbb{Z}\}=\{\dots,-12,-8,-4,0,4,8,12,\dots\}. but I do not understand this condition  (24)\subsetneq (a)

    The ideals (2) (4) and (6) satisfy the condition  (24)\subsetneq (a) ?
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  4. #4
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    Re: Find ideals

    This is in your original
    Quote Originally Posted by cristianoceli View Post
    (24)\subsetneq (a) ( \subsetneq means is a proper subsets of)
    Now you post:
    Quote Originally Posted by cristianoceli View Post
    Means r \in R , Second for example (4)=\{mr:\,r\in\mathbb{Z}\}=\{\dots,-12,-8,-4,0,4,8,12,\dots\}. but I do not understand this condition  (24)\subsetneq (a)
    The ideals (2) (4) and (6) satisfy the condition  (24)\subsetneq (a) ?
    So we are rightly confused as to what you do understand. Do you see why that is?

    Surely you are not saying that you do not understand the notation: $B\subsetneq A~?$
    If you do not understand the notation it means: $[B\subset A] \wedge[\exists x\in A$ such that $x\notin B]$

    If you know what $(24)$ is then is it true that $(24)\subsetneq (4)~?$
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  5. #5
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    Re: Find ideals

    Quote Originally Posted by cristianoceli View Post
    Means r \in R , Second for example (4)=\{mr:\,r\in\mathbb{Z}\}=\{\dots,-12,-8,-4,0,4,8,12,\dots\}.
    You mean (4)= \{4r: r\in\mathbb{Z}\}. But yes, that is \{\dots,-12,-8,-4,0,4,8,12,\dots\}, all multiples of 4. Those dots of course, mean that it continues in that way- it also contains -24= 4(-6), 24= 4(6), -48= 4(-12), 48= 4(12), etc.

    but I do not understand this condition  (24)\subsetneq (a)
    (24)= \{\dots, -48, -24, 24, 48, \dots\} consists of all multiples of 24 which are all multiples of 4 because 24 itself is a multiple of 4.

    Similarly, (6)= \{\dots, -30, -24, -18, -12, -6, 0, 6, 12, 18, 24, 30, \dots\} which, again, contains all multiples of 24 because 24 is a multiple of 6.

    The ideals (2) (4) and (6) satisfy the condition  (24)\subsetneq (a) ?
    Last edited by HallsofIvy; Jul 16th 2017 at 02:19 PM.
    Thanks from cristianoceli
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  6. #6
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    Re: Find ideals

    Quote Originally Posted by Plato View Post

    If you know what $(24)$ is then is it true that $(24)\subsetneq (4)~?$
    Yes, [(24)\subset (4)] y for example  8 \in (4) but  8 \not\in{(24)}. Therefore  [\exists x\in (4) such that  x\notin (24)
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  7. #7
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    Re: Find ideals

    You have it backwards. (24) is a subset of (4). What your example shows is that (4) is not a subset of (24).
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