Is <S> equal to the intersection of all ideals in R that contain S?

[cchatham]

Homework Statement

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]r_is_i: is any integer from 1 to n, r_i[tex]\in[/tex]R [tex]\forall[/tex]i and s_i[tex]\in[/tex]S [tex]\forall[/tex]i} = K

Homework Equations

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 r_i[tex]\in[/tex]R, s_i[tex]\in[/tex]S, each r_is_i[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.

 

[aPhilosopher]

[tex]S \subseteq I \Rightarrow \forall s \in S, (s) \subseteq I[/tex]