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Lets generalize.
A nonempty subset I of
is called an ideal iff (i) and (ii) .Note that condition (i) just says that I is an additive subgroup of
. Any ideal I contains 0 because if (by definition I is nonempty), . Also, if , then .Theorem: I is an ideal of if and only if there exists an integer d such that .
We shall write
to denote the set .Proof: Verifying that
is an ideal is straightforward.Conversely, suppose I is an ideal. If
, then . Otherwise I must contain some positive integers, for if , then and one of and is positive. Then let d be the smallest positive member of I.Since
for all , .On the other hand, if
, we can write where and . But then and so, by minimality of d, we must have . Hence .The above theorem is a particular case of a more general theorem, which is proved in an identical way to the above.
More general theorem: Every Euclidean domain is a principal-ideal domain.
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