Characteristic of Finite Ring is Non-Zero
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Theorem
Let $\struct {R, +, \circ}$ be a finite ring with unity.
Then the characteristic of $R$ is not zero.
Proof
We have that $\struct {R, +, \circ}$ is finite, so its additive group $\struct {R, +}$ is likewise finite.
The result follows by Element of Finite Group is of Finite Order and the definition of characteristic.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Theorem $29$