TITLE: Characteristic of a simple ring is either prime or $0$
QUESTION [10 upvotes]: If $R$ is a simple ring, prove that characteristic of $R$ is either prime or $0$.

Note: in the question the presence of unity is not mentioned, can we do without it?

REPLY [14 votes]: The characteristic of $R$ is the least positive integer $k$ such that $kr=0$, for all $r\in R$, or the characteristic is $0$ if such an integer doesn't exist.
For an integer $m>0$, the set $mR=\{mr:r\in R\}$ is clearly an ideal of $R$. Since $R$ is simple, either $mR=\{0\}$ or $mR=R$. If $k>0$ is the characteristic of $R$, then, by definition, $kR=\{0\}$ and $mR\ne\{0\}$, for $0<m<k$.
Let $k=mn$ be composite, with $m>1$ and $n>1$. Then $mR\ne\{0\}$ and $nR\ne\{0\}$, so $nR=R$; therefore $kR=mnR=m(nR)=mR\ne\{0\}$, a contradiction.