## Integral Domains

**Point of Post: **In this post we discuss the notion of integral domains.

*Motivation*

Intuitively, integral domains are just commutative rings where you can’t multiply non-zero elements of the ring and get back zero. In this post we’ll define this more rigorously and prove that every finite integral domain is a field.

*Integral Domain*

Let be a unital commutative ring with . We say that is an *integral domain *if it has no zero divisors. In other words, has the property that implies that or . We now give some different characterizations of this:

**Theorem: ***Let be a commutative unital ring with . Then, is an integral domain if and only if implies for every and implies for *

**Proof: **Suppose first that is an integral domain. The fact that implies that or that . But, since is an integral domain we know that or and since we have that or that . The fact that implies that is done similarly.

Conversely, suppose that implies for every . Suppose further that for . If we have that and so . Similarly, if then and so . Either way, we’re done$.

If a ring has the property that implies for non-zero we call the ring *left cancellative. *If the analogous property holds on the right we call the ring *right cancellative.* If both of these hold we call the ring just *cancellative*. Thus, in light of this theorem we see that an integral domain is a unital commuative cancellative ring with .

Note though the clear fact that a ring is cancellative if and only if the maps and are injective for every .

With this in mind we can now prove that every finite integral domain is a field.

**Theorem: ***Every finite integral domain is a field.*

**Proof: **Since is already a commutative unital ring with it suffices to show that it is a divison ring. To do this we let be arbitrary and consider the map . By the above remark we know that is injective, and since is finite we may appeal to a common set-theoretic fact to conclude that is surjective. Thus, there exists such that and since is commutative we know that . Thus, since was arbitrary the conclusion follows.

So what’s the quintessential example of a commutative unital ring with which is not an integral domain? Consider , the set of all continuous functions . This is evidently a ring with the usual addition and multiplication of functions. Moreover, it’s clear that it’s commutative and unital with being the constant function . That said, the functions (where is the indicator function on ) and are evidently non-zero elements of yet it’s easy to see that .

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

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