Ring Homomorphism

A ring homomorphisms between $A,B:\mathrm{Ring}$ is a function $f:A\to B$ that preserves addition, multiplication and neutral element.

There is a unique ring homomorphisms from $\mathbb{Z}$ to any ring.

There is also a ring homomorphsm from ${\varphi }_a:\mathbb{Z}[X]\to A$ that maps a polynomial to the result of applying the polynomial to $a$.

Kernel

The kernel of a ring homomorhism is the subset of the domain mapped to zero.