I know that a prime ring R is a ring in which the zero ideal is a prime ideal, i.e., aRb=0 implies either a=0 or b=0. To prove R is 2-torsion free I want to prove that there is no nonzero x in R such that 2x=0. My approach: Suppose there exists a nonzero x in R such that 2x=0. I seek a contradiction but I am not able to find it.
Please someone suggest to me how I should approach this.
Assume that $x\ne0$ but $2x=0$. For every $y\in R$, we have $xR(2y)=(2x)Ry=0$. Since $R$ is prime, it follows that either $x=0$ (which is false by assumption) or $2y=0$. Thus, $2y=0$ for every $y\in R$; i.e., $R$ has characteristic $2$.