Re: proper ideals of Z

crossedproduct <supermanifold@xxxxxxxxx> wrote:

To show that every proper ideal of Z is of the form nZ for some integer n,
we should consider an ideal I for which this is not true, and find elements
c,d in I whose gcd = 1. I'm not sure how to go about doing this.

Since I is nonzero it contains a least element d > 1.
Since I isn't principal, d doesn't divide some c in I.
Hence I contains c mod d = 1 (by leastness of d). QED

Essentially the same proof suffices to show that a
Euclidean domain is a PID. For this one chooses d
of minimal value among all nonzero nonunits of I
(see also the concept of "universal side divisor").

However, it is simpler to avoid this notion and
proceed directly, showing that every nonzero ideal
is generated by any elt of nonzero minimal value.

--Bill Dubuque
.