The first part of the problem is as follows:(adsbygoogle = window.adsbygoogle || []).push({});

Any nonempty set of integers J that fulfills the following two conditions is called an integral ideal:

i) if n and m are in J, then n+m and n-m are in J; and

ii) if n is in J and r is an integer, then rn is in J.

Let Jm be the set of all integers that are integral multiples of a particular integer m. Prove that Jm is an integral ideal.

Second part:

Prove that every integral ideal J is identical with Jm for some m. (Hint: Prove that if J [tex]\neq[/tex]{0}=J_{0}, then there exists positive integers in J. By the least-integer principles, there is a least positive integer in J, say m. Then prove that J=Jm

For the first part, I'm not really sure if there is not much to it, or maybe I'm just missing something. Either way, I am not really sure how to put it down on paper. If m is in J, and Jm is simply some integer multiple of m, rm. Then it automatically satisfies the ii. For part i, n and m can be factored into (r_{1}+r_{2})m or (r_{1}-r_{2})m. Either way this is still an integer multiple of m, which is in J. Not really sure if this is right though...

For the second part I am a bit more lost. How do I prove that there are positive integers in J? When you go to prove that J=Jm, I think you have to use Euclid's division lemma and the concept that the gcd will divide every integer multiple of J. Still not really sure how this all ties in though...

Any help would be greatly appreciated! Thanks!

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# Homework Help: Elementary Number Theory Proof, Integral Ideals

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