Finite Ring with Multiplicative Norm is Field
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Theorem
Let $R$ be a finite ring with a multiplicative norm.
Then $R$ is a field.
Proof
From Ring with Multiplicative Norm has No Proper Zero Divisors, $R$ has no proper zero divisors.
From Finite Ring with No Proper Zero Divisors is Field, $R$ is a field.
$\blacksquare$