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.