The characteristic of a ring with unity is defined to be the least positive integer $n$ such that $1$ plus itself $n$ times $=0$. How does this make sense? $1$ plus itself $n$ times $=n1=n=0$, but $n$ is defined to be nonzero.
One exercise that is bothering me is:
Let $A$ be a finite integral domain. Prove: Let $a$ be any nonzero element of $A$. If $na=0$, where $n\neq0$, then $n$ is a multiple of the characteristic of $A$.
This doesn't make sense. If $A$ is an integral domain, and $a$ is nonzero, and if $na=0$ where $n\neq0$, then this statement doesn't make sense. The characteristic is defined to be nonzero, and if $n$ is multiple of the characteristic then it is nonzero, and also $a$ is nonzero, but $na=0$. This contradicts being an integral domain.
Any help?
The abstract answer to your question involves the (unique!) ring homomorphism $\mathbb Z\to R$, which exists if $R$ itself has a unit element $1_R$. This homomorphism sends the natural number $1$ to $1_R$, and a positive number $n\in\mathbb Z$ to the result of adding $1_R$ to itself $n$ times. Any homomorphism has a kernel, in this case it’ll be an ideal of the domain $\mathbb Z$, and so generated by a well-defined nonnegative integer $\chi$. This integer is the characteristic of $R$, and it may be any nonnegative integer. Looking through the construction and definition, you see that that characteristic is the smallest positive $n$ such that adding $1_r$ to itself $n$ times gives a result of zero; but if there is no such positive number, the construction gives $\chi=0$.