Characteristic times Ring Element is Ring Zero

Theorem

Let $\struct {R, +, \circ}$ be a ring with unity.

Let the zero of $R$ be $0_R$ and the unity of $R$ be $1_R$.

Let the characteristic of $R$ be $n$.

Then:

$\forall a \in R: n \cdot a = 0_R$

Proof

If $a = 0_R$ then $n \cdot a = 0_R$ is immediate.

So let $a \in R: a \ne 0_R$.

Then:

\(\ds n \cdot a\)

\(=\)

\(\ds n \cdot \paren {1_R \circ a}\)

Definition of Unity of Ring

\(\ds \)

\(=\)

\(\ds \paren {n \cdot 1_R} \circ a\)

Integral Multiple of Ring Element

\(\ds \)

\(=\)

\(\ds 0_R \circ a\)

Definition of Characteristic of Ring

\(\ds \)

\(=\)

\(\ds 0_R\)

Definition of Ring Zero

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Theorem $28$