Constant Mapping to Identity is Homomorphism/Rings

Theorem
Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings with zeroes $0_1$ and $0_2$ respectively.

Let $\zeta$ be the zero homomorphism from $R_1$ to $R_2$, that is:
 * $\forall x \in R_1: \map \zeta x = 0_2$

Then $\zeta$ is a ring homomorphism whose image is $\set {0_2}$ and whose kernel is $R_1$.

Proof
The additive groups of $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ are $\struct {R_1, +_1}$ and $\struct {R_2, +_2}$ respectively.

Their identities are $0_1$ and $0_2$ respectively.

Thus from the Constant Mapping to Group Identity is Homomorphism we have that $\zeta: \struct {R_1, +_1} \to \struct {R_2, +_2}$ is a group homomorphism:
 * $\map \zeta {x +_1 y} = \map \zeta x +_2 \map \zeta y$

Then we have:

The results about image and kernel follow directly by definition.