Characteristic of Finite Ring is Non-Zero

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 $29$