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Ideals and Commutative Rings

  1. Sep 28, 2009 #1
    1. The problem statement, all variables and given/known data
    The following are equivalent for S[tex]\subseteq[/tex]R, S[tex]\neq[/tex][tex]\oslash[/tex], and R is a commutative ring with unity(multiplicative identity):

    1. <S> is the ideal generated by S.
    2. <S> = [tex]\bigcap[/tex](I Ideal in R, S[tex]\subseteq[/tex]I) = J
    3. <S> = {[tex]\sum[/tex]risi: is any integer from 1 to n, ri[tex]\in[/tex]R [tex]\forall[/tex]i and si[tex]\in[/tex]S [tex]\forall[/tex]i} = K

    2. Relevant equations
    3. The attempt at a solution
    It's been some time since I worked on this and at the time I understood everything I was working on but now when I look at it, I'm thoroughly confused. Where I got stuck is showing 2 [tex]\Rightarrow[/tex] 3. I've got, assume <S> = J. Choose a [tex]\in[/tex]K. Let I be an ideal of R that contains S. Because each ri[tex]\in[/tex]R, si[tex]\in[/tex]S, each risi[tex]\in[/tex]I by IO closure. Then a [tex]\in[/tex] I by closure under addition. Thus a [tex]\in[/tex] J and K[tex]\subseteq[/tex]J.

    I'm having trouble with starting to show that J[tex]\subseteq[/tex]K.
     
  2. jcsd
  3. Sep 28, 2009 #2
    [tex]S \subseteq I \Rightarrow \forall s \in S, (s) \subseteq I[/tex]
     
    Last edited: Sep 28, 2009
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