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Math Help - Question about Ideals.

  1. #1
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    Question about Ideals.

    Suppose that R is a commutative ring and let I be an ideal of R. Suppose
    that r,s\in R are such that there are positive integers k,l with r^k,s^l\in I.
    I basically need to know if it's true that r^k,s^l\in I\implies r,s\in I, if not, give a counter example.

    My working:

    Given the definition of Ideals, then a\in I, r\in R \implies ar=ra\in I

    I don't think the statement is true, since the definition isn't an if and only if implication, i.e it isn't necessarily true that given r\in R, r^2=rr\in I \implies r\in I letting a=r, k=2

    Can someone please help me think of a counter example, or show me that i'm wrong. Thank you.
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  2. #2
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    In definitions when authors says if they mean if and only if.

    Since everybody know therefore they write if insted of if and only if.
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  3. #3
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    Quote Originally Posted by skamoni View Post
    Suppose that R is a commutative ring and let I be an ideal of R. Suppose
    that r,s\in R are such that there are positive integers k,l with r^k,s^l\in I.
    I basically need to know if it's true that r^k,s^l\in I\implies r,s\in I, if not, give a counter example.

    My working:

    Given the definition of Ideals, then a\in I, r\in R \implies ar=ra\in I

    I don't think the statement is true, since the definition isn't an if and only if implication, i.e it isn't necessarily true that given r\in R, r^2=rr\in I \implies r\in I letting a=r, k=2

    Can someone please help me think of a counter example, or show me that i'm wrong. Thank you.
    The set of all multiples of 4 is an ideal in the ring of integers. It contains 2^2 but it does not contain 2.
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