What is the characteristic of a ring with the property that if $mx = my$, then $x = y$? Suppose $x,y \in R$ where $R$ is a ring (not necessarily with an identity) and suppose $m$ is a positive integer.   What is the characteristic of a ring with the property that if $mx = my$, then $x = y$?
This is not true if the characteristic divides $m$.
If the characteristic has a factor in common with $m$, then in some rings of integers modulo $n$ this would not be true either for every $x,y$.
So is it only true in characteristic 0 rings?

If this property holds and the ring has characteristic $c$, then in particular we must have either $c=0$ or $\gcd(m,c)=1$. Otherwise, let $x = \frac{c}{\gcd(m,c)}a$ for any element $a$ such that $\frac{c}{\gcd(m,c)}a \ne 0$ (this must exist, because $\frac{c}{\gcd(m,c)} < c$), and let $y=0$. Then $my = mx = 0$, but $x \ne y$.
Also, whenever $m = \pm 1$, this property is guaranteed to hold for any ring; when $m=0$, it only holds for the trivial ring.
However, for any other $m$, there are examples of any characteristic $c \ne 1$ where it does not hold. For example, take $\mathbb Z[x]/(mx, c)$. Here, $mx=0$ but $x \ne 0$.
In conclusion, given this property, we can limit the characteristic of the ring; however, the characteristic does not help us determine if this property holds.
