Showing that Every Ideal of R has the Form mR

[kimberu]

Homework Statement

If R = Z_n, show that every ideal of R has the form mR for some integer m.

Homework Equations

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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!

 

[VeeEight]

Use the division algorithm.

 

[kimberu]

Use the division algorithm.

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

 

[TMM]

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

 

[VeeEight]

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

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.

 

[kimberu]

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

Would it be (m+n)R?

 

[kimberu]

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.

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?

 

[VeeEight]

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

 

[kimberu]

Yes.

Thank you so much for walking me through! :)

 

[VeeEight]

No problem, cheers.