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In this wikipedia article it is said that set Z is a principal ideal domain, i.e. each one of its ideals can be generated by a single element. But if we consider set C of all composite integer numbers (Z without primes and 0), wouldn't it be an ideal? If we take arbitrary element from Z and take a product of a composite number and an arbitrary integer we will get a composite, thus an element of C? And, as far as I can understand, C can not be generated by a single element. So it is not a principle ideal.

I would appreciate pointing to my mistake very much.

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    Thanks to all of you guys! I got it now.2011-06-21

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The problem is that the set of all composite integers does not form an ideal. For example if you add $21 - 4$ you get $17$ which is a prime and thus not composite. That's why there's no contradiction.

Remember that for a subset of a ring to be an ideal it must be closed under addition and under taking multiples by elements of the ring, and in this case the set of all composite integers is not closed under addition.

And the fact that $\mathbb{Z}$ is a principal ideal domain is because you have division with remainder in the set of integers.

It is a more general fact that any ring that has this sort of division, which is called an Euclidean Domain, is as a consequence a Principal Ideal Domain.

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    @ikostia No problem, I'm glad it helped you.2011-06-21