1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook