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Math Help - The Nature of Principal Ideals

  1. #1
    Super Member Bernhard's Avatar
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    The Nature of Principal Ideals

    Fraleigh (A First Course in Abstract Algebra) defines principal ideals in section 27 on page 250. His definition is as follows:

    ================================================== =============================================

    "27.21 Definition

    If R is a commutative ring with unity and  a \in R , the ideal  \{ ra | r \in R \} of all multiples of a is the principal ideal generated by a and is denoted <a>.

    An ideal N of R is a principal ideal if N = <a> for some  a \in R

    ================================================== ===============================================

    Consider  N =\{ ra | r \in R \} ...........................(1)

    If we take r = a in (1) then we have  ra = aa = a^2 \in N

    If we take r = a and  a^2 \in N the we have using (1) again that  ra = a^2 a = a^3 \in  N

    Continuing this, then we have  a, a^2, a^3, a^4, a^5 , .... all belonging to N along with the other elements where  r \ne a

    Is the above analysis correct regarding the nature of principal ideals?

    Would really appreciate this issue being clarified.

    Peter
    Last edited by Bernhard; March 15th 2013 at 06:26 AM.
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  2. #2
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    Re: The Nature of Principal Ideals

    Yes. An ideal is also closed under addition (in fact, it is a subgroup of (R,+)), so you also get elements such as a^2+ a^7 +ra+sa with  r,s\in R. However, you can see that this is the same as (a+a^6+r+s)a, where (a+a^6+r+s)\in R. Since multiplication by an element in R and adding elements in N generates the whole ideal, this shows (by example, not proof) that \langle a \rangle = \{ra|r\in R\}
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  3. #3
    Super Member Bernhard's Avatar
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    Re: The Nature of Principal Ideals

    Thanks Gusbob ... That post was really helpful in characterizing the other elements you mentioned!!!

    Appreciate your help

    Peter
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