# Identity Mapping is Automorphism/Rings

< Identity Mapping is Automorphism(Redirected from Identity Mapping is Ring Automorphism)

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## Theorem

Let $\struct {R, +, \circ}$ be a ring whose zero is $0$.

Then $I_R: \struct {R, +, \circ} \to \struct {R, +, \circ}$ is a ring automorphism.

Its kernel is $\set 0$.

## Proof

The result Identity Mapping is Automorphism holds directly, for both $+$ and $\circ$.

As $I_R$ is a bijection, the only element that maps to $0$ is $0$ itself.

Thus the kernel is $\set 0$.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 24$. Homomorphisms: Example $43$