[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
- is an integral domain.
- Any field is an integral domain. (The proof is an exercise.)
%%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 . %%
- When is a prime integer, the ring of integers mod is an integral domain.
- When is a [composite_number composite] integer, the ring is not an integral domain. Indeed, if with positive integers, then in .
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 . %%