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

[cchatham]

Homework Statement

The following are equivalent for S\subseteqR, S\neq\oslash, and R is a commutative ring with unity(multiplicative identity):

1. <S> is the ideal generated by S.
2. <S> = \bigcap(I Ideal in R, S\subseteqI) = J
3. <S> = {\sumr_is_i: is any integer from 1 to n, r_i\inR \foralli and s_i\inS \foralli} = 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 \Rightarrow 3. I've got, assume <S> = J. Choose a \inK. Let I be an ideal of R that contains S. Because each r_i\inR, s_i\inS, each r_is_i\inI by IO closure. Then a \in I by closure under addition. Thus a \in J and K\subseteqJ.

I'm having trouble with starting to show that J\subseteqK.

 

[aPhilosopher]

S \subseteq I \Rightarrow \forall s \in S, (s) \subseteq I