If $F$ is a field, then it is a well-known result that the following are equivalent for two norms $|\cdot|_1$ and $|\cdot|_2$ on $F$:
Any sequence in $F$ is Cauchy with respect to $|\cdot|_1$ if and only if it is Cauchy with respect to $|\cdot|_2$.
There exists some $\alpha > 0$ such that $|x|_1 = |x|_2^\alpha$ for all $x\in F$.
Is this still true in some sense for norms on rings? If so, could anyone give me a reference? I think the standard proof uses division at a couple of steps.
Just in case, from a norm on a ring I require that
- $|x| = 0 \iff x = 0$.
- $|xy| = |x|\cdot|y|$ (in particular, there are no zero divisors).
- $|x+y| \le |x| + |y|$.