characteristic of ${\mathbb Z_2 }$ is $0$?

Question

I tried to solve the characteristic of ${ \mathbb Z_p }$ = $0$ or prime $p$ .

In ${ \mathbb Z_2 },$ nothing gives me $0$ where $nr = 0$ and $n \ne 0,$ where ${ n , r \in R }$

Is my solution correct?

Answer 1

No, the characteristic of $\mathbb Z/n\mathbb Z$ is $n$; in particular, the characteristic of $\mathbb Z/2\mathbb Z$ is $2$.

The characteristic of a ring is the smallest $n\in\mathbb N$ such that $n1=\underbrace{1+...+1}_{n \text{ times}}=0,$

where $1$ is the multiplicative identity and $0$ is the additive identity in the ring.

(If no such $n$ exists, the characteristic is $0$.)

You might have been confused because you were thinking of $n$ in the ring, not in $\mathbb N$.

Answer 2

Characteristic is not in the set, it is from positive integers. If there is not a positive integer n such that n.a=0 for all a element of R, then char R=0