Identity Mapping is Automorphism/Rings

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 $43$
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