Definition:Zero Homomorphism

Definition

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

Consider the mapping $\zeta: R_1 \to R_2$ defined as:

$\forall r \in R_1: \map \zeta r = 0_2$

Then $\zeta$ is the zero homomorphism from $R_1$ to $R_2$.

Also known as

The zero homomorphism is also referred to by some authors as the trivial homomorphism.

Also see

• Constant Mapping to Identity is Homomorphism: $\zeta$ is indeed a (ring) homomorphism.

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms: Example $1$