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# orders of elements in integral domain

###### Theorem.

Let $(D,\,+,\,\cdot)$ be an integral domain, i.e. a commutative ring with non-zero unity 1 and no zero divisors. All non-zero elements of $D$ have the same order in the additive group $(D,\,+)$.

Proof. Let $a$ be arbitrary non-zero element. Any multiple $na$ may be written as

$na=n(1a)=\underbrace{1a+1a+\cdots+1a}_{{n}}=(\underbrace{1+1+\cdots+1}_{{n}})a% =(n1)a.$ |

Thus, because $a\neq 0$ and there are no zero divisors, an equation $na=0$ is equivalent with the equation $n1=0$. So $a$ must have the same order as the unity of $D$.

Note. The order of the unity element is the characteristic of the integral domain, which is 0 or a positive prime number.

Related:

OrderGroup, IdealOfElementsWithFiniteOrder

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

13G05*no label found*

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