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Math Help - Principal ideal in a ring without identity

  1. #1
    Super Member Bernhard's Avatar
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    Principal ideal in a ring without identity

    (Hungerford exercise 31, page 143)

    Let R be a commutative ring without identity and let  a \in R

    Show that  A = \{ ra + na \  | \  r \in R, n \in \mathbb{Z} \} is an ideal containing a and that every ideal containing a also contains A. (A is called the prinicipal ideal generated by a)
    Last edited by Bernhard; March 15th 2013 at 04:08 PM.
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  2. #2
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    Re: Principal ideal in a ring without identity

    For the first one: take r=0, n=1

    For the second one: In an ideal I, multiplication by elements in R and addition by elements in I produces elements in I. The result should now be obvious (though you should prove it more rigorously, shouldn't be more than a few lines).
    Thanks from Bernhard
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