# Definition:Zero Homomorphism

## Contents

## 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 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

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

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.3$: Example $1$