Ideals with subsets and divides

[phyguy321]

Homework Statement

Let I = <f(x)>, J =<g(x)> be ideals in F[x]. prove that I$$\subset$$J $$\leftrightarrow$$ g(x)|f(x)

Homework Equations

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

 

[e(ho0n3]

If I is a subset of J then does that mean that f is in J

Correct.

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

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

 

[phyguy321]

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

 

[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).

 

[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)?

 

[e(ho0n3]

Correct.