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Homework Help: What is the ideal generated by a and b.

  1. Sep 3, 2008 #1
    what is "the ideal generated by a and b."

    I'm doing a problem and I'm a little confused on the notation.

    Let R be a ring and [tex]a,b \in R[/tex]. Prove that [tex](a, b)[/tex], the ideal generated by a and b is the...

    I want to do this on my own so I left out the rest of the problem. I just need to know if [tex](a, b) = aR \cup bR[/tex]. Or is it some kind of set of ordered pairs?
     
  2. jcsd
  3. Sep 3, 2008 #2

    Dick

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    Re: what is "the ideal generated by a and b."

    It's not the union aR and bR. It's the set of all sums of the form aR+bR. They aren't the same. And unless (a,b) is some funny notation different from {a,b} then it has nothing to do with ordered pairs.
     
  4. Sep 3, 2008 #3
    Re: what is "the ideal generated by a and b."

    Okay that makes sense. The way I described it it wouldn't even be closed under addition.
     
  5. Sep 5, 2008 #4

    morphism

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    Re: what is "the ideal generated by a and b."

    General remark: Usually the phrase "object of type T generated by a set X" means "smallest object of type T containing the set X." And usually this is the 'intersection' (or appropriate analogue) of all objects of type T containing the set X.
     
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