# Ideals and Commutative Rings

1. Sep 28, 2009

### cchatham

1. The problem statement, all variables and given/known data
The following are equivalent for S$$\subseteq$$R, 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$$\subseteq$$I) = J
3. <S> = {$$\sum$$risi: is any integer from 1 to n, ri$$\in$$R $$\forall$$i and si$$\in$$S $$\forall$$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 $$\Rightarrow$$ 3. I've got, assume <S> = J. Choose a $$\in$$K. Let I be an ideal of R that contains S. Because each ri$$\in$$R, si$$\in$$S, each risi$$\in$$I by IO closure. Then a $$\in$$ I by closure under addition. Thus a $$\in$$ J and K$$\subseteq$$J.

I'm having trouble with starting to show that J$$\subseteq$$K.

2. Sep 28, 2009

### aPhilosopher

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

Last edited: Sep 28, 2009