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I Two-sided Prinicipal Ideal - the Noncommutative Case - D&F

  1. Aug 20, 2016 #1
    I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 7.4 Properties of Ideals ... ...

    I have a basic question regarding the generation of a two sided principal ideal in the noncommutative case ...

    In Section 7.4 on pages 251-252 Dummit and Foote write the following:



    ?temp_hash=ca92414d4802ac8f6cfd56bd2273af5e.png
    ?temp_hash=ca92414d4802ac8f6cfd56bd2273af5e.png


    In the above text we read:


    " ... ... If ##R## is not commutative, however, the set ##\{ ras \ | \ r, s \in R \}## is not necessarily the two-sided ideal generated by ##a## since it need not be closed under addition (in this case the ideal generated by ##a## is the ideal ##RaR##, which consists of all finite sums of elements of the form ##ras, r,s \in R##). ... ... "



    I must confess I do not understand or follow this argument ... I hope someone can clarify (slowly and clearly confused.png ) what it means ... ...


    Specifically ... ... why, exactly, is the set ##\{ ras \ | \ r, s \in R \}## not necessarily the two-sided ideal generated by ##a##?


    The reason given is "since it need not be closed under addition" ... I definitely do not follow this statement ... surely an ideal is closed under addition! ...


    ... ... and why, exactly does the two-sided ideal generated by a consist of all finite sums of elements of the form ##ras, r,s \in R## ... ... ?



    Hope someone can help ... ...

    Peter


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


    To give readers the background and context to the above text from Dummit and Foote, I am providing the introductory page of Section 7.4 as follows ... ...


    ?temp_hash=ca92414d4802ac8f6cfd56bd2273af5e.png
     

    Attached Files:

  2. jcsd
  3. Aug 20, 2016 #2

    fresh_42

    Staff: Mentor

    In an ideal ##\mathcal{I}## you have ##x+y\in \mathcal{I}## for elements ##x\,,\,y\in \mathcal{I}##. This is a basic part of its definition.
    If you now have two elements ##ras \; (r,s\in R\,; a\in \mathcal{I})## and ##paq \; (p,q \in R\,; a\in \mathcal{I})## there is - in general - no way to write ##ras+paq=uav## because you cannot pull the factors ##r,s,p,q## on the other side of ##a##.
    Therefore ##\mathcal{I}=(a) = RaR = LC(\{ras \,|\, r,s \in R\}) \supsetneq \{ras \,|\, r,s \in R\}## is - in general - a proper inclusion. The latter is only a set.

    In the commutative case we have
    ##ras+paq=r(as)+p(aq)=r(sa)+p(qa)=(rs)a+(pq)a=(rs+pq)a## and all linear combinations are of the form ##ua\;(u=rs+pq \in R)##.
    However, this calculation is not allowed in non-commutative rings.
     
  4. Aug 20, 2016 #3

    micromass

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    You should try to find a counterexamples yourself where ##\{ras~\vert~r,s\in R\}## is not an ideal. Think of some simple noncommutative rings.
     
  5. Aug 21, 2016 #4
    Thanks fresh_42 ... that was most helpful ...

    Thanks for suggestion micromass ... will try messing around with 2 by 2 matrices ...

    Peter
     
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