### Integral domains and characteristics

Today we shall talk about the characteristic of an integral domain, concentrating mainly on misconceptions and important points.

An integral domain is a commutative ring with the property that if and , then . Hence, if , then or (or both).

The characteristic of an integral domain is the lowest positive integer such that .

Let . Then . This is because .

If , then we have . This is obvious for . If , then this implies , which contradicts the fact that is the lowest positive integer such that added times to itself is equal to . Hence, if is the characteristic of the integral domain , then it is the lowest positive integer such that any non-zero member of , added times to itself, gives . No member of can be added a lower number of times to itself to give .

Sometimes is written as . One should remember that this has nothing to do the multiplication operator in the ring. In other words, this does not imply that , where is a member of the domain. In fact, does NOT have to be a member of the domain. It is just an arbitrary positive integer.

Now on to an important point: something that is not emphasized, but should be. Any expression of the form

.

Now use this knowledge to prove that the characteristic of an integral domain, if finite, has to be or prime.