Ring with Multiplicative Norm has No Proper Zero Divisors

Theorem
Let $\struct {R, +, \circ}$ 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



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

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.