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

    - Definition


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




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

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

    First, you mean "$\displaystyle r\in R$" rather than "$\displaystyle r\in A$", 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 $\displaystyle r \in R$ , Second for example $\displaystyle (4)=\{mr:\,r\in\mathbb{Z}\}=\{\dots,-12,-8,-4,0,4,8,12,\dots\}.$ but I do not understand this condition $\displaystyle (24)\subsetneq (a)$

    The ideals (2) (4) and (6) satisfy the condition $\displaystyle (24)\subsetneq (a)$ ?
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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
    $\displaystyle (24)\subsetneq (a) $ ($\displaystyle \subsetneq$ means is a proper subsets of)
    Now you post:
    Quote Originally Posted by cristianoceli View Post
    Means $\displaystyle r \in R$ , Second for example $\displaystyle (4)=\{mr:\,r\in\mathbb{Z}\}=\{\dots,-12,-8,-4,0,4,8,12,\dots\}.$ but I do not understand this condition $\displaystyle (24)\subsetneq (a)$
    The ideals (2) (4) and (6) satisfy the condition $\displaystyle (24)\subsetneq (a)$ ?
    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 $\displaystyle r \in R$ , Second for example $\displaystyle (4)=\{mr:\,r\in\mathbb{Z}\}=\{\dots,-12,-8,-4,0,4,8,12,\dots\}.$
    You mean $\displaystyle (4)= \{4r: r\in\mathbb{Z}\}$. But yes, that is $\displaystyle \{\dots,-12,-8,-4,0,4,8,12,\dots\}$, 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 $\displaystyle (24)\subsetneq (a)$
    $\displaystyle (24)= \{\dots, -48, -24, 24, 48, \dots\}$ consists of all multiples of 24 which are all multiples of 4 because 24 itself is a multiple of 4.

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

    The ideals (2) (4) and (6) satisfy the condition $\displaystyle (24)\subsetneq (a)$ ?
    Last edited by HallsofIvy; Jul 16th 2017 at 02:19 PM.
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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, $\displaystyle [(24)\subset (4)] $ y for example $\displaystyle 8 \in (4) $ but $\displaystyle 8 \not\in{(24)}$. Therefore $\displaystyle [\exists x\in (4)$ such that $\displaystyle x\notin (24)$
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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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