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.