Characteristic of a ring

Builds on Ring

Definition

Let $R$ be a ring. For a positive integer $n$ , write $n \cdot 1 = 1 + 1 + \dots + 1$ , with $n$ terms. If there is a positive integer $n$ with $n \cdot 1 = 0$ in $R$ , then the characteristic of $R$ is the least such $n$ . If there is no such $n$ , then the characteristic of $R$ is $0$ .

Remarks

The rings $\mathbb{Z}$ , $\mathbb{Q}$ , $\mathbb{R}$ and $\mathbb{C}$ have characteristic $0$ . For $n \ge 1$ , the ring $\mathbb{Z}/n\mathbb{Z}$ has characteristic $n$ . The characteristic of $R$ is the nonnegative generator of the kernel of the unique ring homomorphism $\mathbb{Z} \to R$ . The characteristic of an integral domain is zero or prime.

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Group concept
Ring definition
Characteristic of a ring definition

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Tag	`0P7CH1CND5`
URI	`m2://0P7CH1CND5`
Kind	definition
Writing	W1 stated
MSC	13A99
Defines	Characteristic of a ring
Uses	Ring

Characteristic of a ring

Definition

Remarks

Reading path

Where this leads

Further reading

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