# Definition:Zero Homomorphism

(Redirected from Definition:Trivial Homomorphism)

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## 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 $1$