Integral domain

https://arbital.com/p/integral_domain

by Patrick Stevens Jul 28 2016

An integral domain is a ring where the only way to express zero as a product is by having zero as one of the terms.


[summary: An integral domain is a ring in which the only way to make as a product is to multiply by something. For instance, in an integral domain like [48l ], is not equal to because neither nor is.]

[summary(Technical): An integral domain is a ring in which implies or . (We exclude the ring with one element: that is conventionally not considered an integral domain.)]

In keeping with ring theory as the attempt to isolate each individual property of [48l ] and work out how the properties interplay with each other, we define the notion of integral domain to capture the fact that if then or . That is, an integral domain is one which has no "zero divisors": cannot be nontrivially expressed as a product. (For uninteresting reasons, we also exclude the ring with one element, in which , from being an integral domain.)

Examples

%%hidden(Show solution): Suppose , but . We wish to show that .

Since we are working in a field, has an inverse ; multiply both sides by to obtain . Simplifying, we obtain . %%

Properties

The reason we care about integral domains is because they are precisely the rings in which we may cancel products: if and then . %%hidden(Proof): Indeed, if then so , and hence (in an integral domain) or .

Moreover, if we are not in an integral domain, say but . Then , but , so we can't cancel the from both sides. %%

Finite integral domains

If a ring is both finite and an integral domain, then it is a field. The proof is an exercise. %%hidden(Show solution): Given , we wish to find a multiplicative inverse.

Since there are only finitely many elements of the ring, consider . This set is a subset of , because the multiplication of is closed. Moreover, every element is distinct, because if then we can cancel the (because we are in an integral domain), so .

Since there are -many elements of the subset (where refers to the Cardinality), and since is finite, must in fact be itself.

Therefore in particular , so for some . %%