Divisibility

There is an interesting preorder we can put on the nonzero elements of any commutative ring with unit. If and are nonzero elements of a ring , we say that divides — and write — if there is an so that . The identity trivially divides every other nonzero element of .

We can easily check that this defines a preorder. Any element divides itself, since . Further, if and then there exist and so that and , so and we have .

On the other hand, this preorder is almost never a partial order. In fact since and we see that and , and most of the time . In general, when both and we say that and are associates. Any unit comes with an inverse , so we have and . If for some unit , then and are associates because .

We can pull a partial order out of this preorder with a little trick that works for any preorder. Given a preorder we write if both and . Then we can check that defines an equivalence relation on , so we can form the set of its equivalence classes. Then descends to an honest partial order on .

One place that divisibility shows up a lot is in the ring of integers. Clearly and are associate. If and are positive integers with , then there is another positive integer so that . If then . Otherwise . Thus the only way two positive integers can be associate is if they are the same. The preorder of divisibility on induces a partial order of divisibility on .