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 $a\neq 0$ 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.

Integral domains and characteristics

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