# Homework Help: Ideals of Rings

1. Apr 22, 2010

### kimberu

1. The problem statement, all variables and given/known data
If R = Zn, show that every ideal of R has the form mR for some integer m.

2. Relevant equations
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3. The attempt at a solution
Well, by a previous problem I showed mR is the principal ideal of the ring, but I don't know if that's relevant. I was given the hint to try to use GCDs somehow, but I really have no ideas.

thanks so much!

2. Apr 22, 2010

### VeeEight

Use the division algorithm.

3. Apr 22, 2010

### kimberu

You mean, say that n = qd + r, or for m?
Sorry, I'm totally lost on this problem.

4. Apr 22, 2010

### TMM

Suppose you have an ideal of the form mR+nR. How can you express this as a principal ideal?

5. Apr 22, 2010

### VeeEight

Let a be the smallest positive element in your ideal I. If you have some element x in I, then x = aq + s for some s less then a.

6. Apr 22, 2010

### kimberu

Would it be (m+n)R?

7. Apr 22, 2010

### kimberu

So from here...can I say that s must equal 0 and x = aq for all x, since otherwise it's a contradiction because a is the smallest element?

8. Apr 22, 2010

### VeeEight

Yes. You can also reproduce this proof for other rings such as the Eisenstein integers and the Gaussian integers.

9. Apr 22, 2010

### kimberu

Thank you so much for walking me through! :)

10. Apr 22, 2010

### VeeEight

No problem, cheers.