Ring Homomorphism

A ring homomorphism is a function $\phi : R \to S$ between rings that preserves both operations and the multiplicative identity: $\phi(a + b) = \phi(a) + \phi(b)$, $\phi(a \cdot b) = \phi(a) \cdot \phi(b)$, and $\phi(1_R) = 1_S$.

The kernel $\ker(\phi) = \{r \in R \mid \phi(r) = 0\}$ is always an ideal of $R$. The first isomorphism theorem for rings states that $R/\ker(\phi)$ is isomorphic to the image of $\phi$, paralleling the corresponding theorem for groups.

In category theory, ring homomorphisms are the morphisms in the category of rings.