Definition:Zero Homomorphism

Definition
Let $\left({R_1, +_1, \circ_1}\right)$ and $\left({R_2, +_2, \circ_2}\right)$ be rings with zeroes $0_1$ and $0_2$ respectively.

Consider the mapping $\zeta: R_1 \to R_2$ defined as:
 * $\forall r \in R_1: \zeta \left({r}\right) = 0_2$

Then $\zeta$ is the zero homomorphism (or the trivial homomorphism) from $R_1$ to $R_2$.

Also see
In Constant Mapping to Identity is Homomorphism it is demonstrated that $\zeta$ is indeed a (ring) homomorphism.