Characteristic times Ring Element is Ring Zero
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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 $28$