# Ideals with subsets and divides

1. Oct 22, 2008

### phyguy321

1. The problem statement, all variables and given/known data
Let I = <f(x)>, J =<g(x)> be ideals in F[x]. prove that I$$\subset$$J $$\leftrightarrow$$ g(x)|f(x)

2. Relevant equations

3. The attempt at a solution
If I is a subset of J then does that mean that f is in J also and by definition of an ideal g*some b in J must equal something in J so g|f? because g|f means that f=bg for some b in J

2. Oct 22, 2008

### e(ho0n3

Correct.

Sort of. I would just use the fact that J is generated by g(x).

3. Oct 23, 2008

### phyguy321

what does it mean that J is generated by g(x)? in layman's terms

4. Oct 23, 2008

### e(ho0n3

It means J = {a(x)g(x) : a(x) in F[x]}, in other words J is the set of all "multiples" of g(x).

5. Oct 23, 2008

### phyguy321

So if f(x) is in J and J = {a(x)g(x): a(x) in F[x]} then f(x) = a(x)g(x) therefore g(x)|f(x)?

6. Oct 23, 2008

Correct.