Integral domains and characteristics

by ayushkhaitan3437

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 a\neq 0 and b\neq 0, then ab\neq 0. Hence, if ab=0, then a=0 or b=0 (or both).

The characteristic of an integral domain is the lowest positive integer c such that \underbrace{1+1+\dots +1}_{\text{ c times}}=0.

Let a\in R. Then \underbrace{a+a+\dots +a}_{\text{ c times}}=a\underbrace{(1+1+\dots +1)}_{\text{ c times}}=0. This is because a.0=0.

If \underbrace{a+a+\dots +a}_{\text{ d<c times}}=0, then we have a\underbrace{(1+1+\dots +1)}_{\text{ d times}}=0. This is obvious for a=0. If a\neq 0, then this implies \underbrace{1+1+\dots +1}_{\text{ d times}}=0, which contradicts the fact that c is the lowest positive integer such that 1 added c times to itself is equal to 0. Hence, if c is the characteristic of the integral domain D, then it is the lowest positive integer such that any non-zero member of D, added c times to itself, gives 0. No member of D can be added a lower number of times to itself to give 0.

Sometimes \underbrace{a+a+\dots +a}_{\text{ c times}} is written as ca. One should remember that this has nothing to do the multiplication operator in the ring. In other words, this does not imply that \underbrace{a+a+\dots +a}_{\text{ c times}}=c.a, where c is a member of the domain. In fact, c 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

\underbrace{\underbrace{a+a+\dots +a}_{\text{m times}}+\underbrace{a+a+\dots +a}_{\text{m times}}+\dots +\underbrace{a+a+\dots +a}_{\text{m times}}}_{\text{n times}}=\underbrace{(a+a+\dots +a)}_{\text{m times}}(\underbrace{1+1+\dots +1}_{\text{n times}}).

Now use this knowledge to prove that the characteristic of an integral domain, if finite, has to be 0 or prime.