Identity Mapping is Automorphism/Rings

Theorem
Let $\left({R, +, \circ}\right)$ be a ring whose zero is $0$.

Then $I_R: \left({R, +, \circ}\right) \to \left({R, +, \circ}\right)$ is a ring automorphism.

Its kernel is $\left\{{0}\right\}$.

Proof
The result Identity Mapping is an 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 $\left\{{0}\right\}$.