Ring with Multiplicative Norm has No Proper Zero Divisors

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Let its zero be denoted by $0_R$.

Let $\norm{\,\cdot\,}$ be a multiplicative norm on $R$.

Then $R$ has no proper zero divisors.

That is:


 * $\forall x, y \in R^*: x \circ y \ne 0_R$

where $R^*$ is defined as $R \setminus \set {0_R}$.

Proof
Assume otherwise:


 * $\exists x, y \in {R^*} : x \circ y = 0_R$

By positive definiteness:
 * $x, y \ne 0_R \iff \norm{x}, \norm{y} \ne 0$

Thus:


 * $\norm{x} \norm{y} \ne 0$

But we also have:

a contradiction.