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 $\set 0$.