# Homomorphism with Cancellable Codomain Preserves Identity

## Contents

## Theorem

Let $\struct{S, \circ}$ and $\struct{T, *}$ be algebraic structures.

Let $\phi: \struct{S, \circ} \to \struct{T, *}$ be a homomorphism.

Let $\struct{S, \circ}$ have an identity $e_S$.

Let $\struct{T, *}$ have an identity $e_T$.

Let every element of $\struct{T, *}$ be cancellable.

Then $\map \phi {e_S}$ is the identity $e_T$.

## Proof

Let $\struct{S, \circ}$ be an algebraic structure in which $\circ$ has an identity $e_S$.

Let $\struct{T, *}$ be an algebraic structure in which $*$ has an identity $e_T$.

Let $\struct{T, *}$ be such that every element is cancellable.

Let $\phi: \struct{S, \circ} \to \struct{T, *}$ be a homomorphism.

Every element of $\struct{T, *}$ is cancellable.

Suppose there is an idempotent element in $\struct{T, *}$

So from Identity is only Idempotent Cancellable Element, it must be the identity $e_T$.

Thus:

\(\displaystyle \map \phi {e_S} * \map \phi {e_S}\) | \(=\) | \(\displaystyle \map \phi {e_S \circ e_S}\) | by the morphism property of $\circ$ under $\phi$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \phi {e_S}\) | as $e_S$ is the identity of $\left({S, \circ}\right)$ |

So $\map \phi {e_S}$ is idempotent in $\struct{T, *}$ and the result follows.

$\blacksquare$

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 12$: Theorem $12.3: \ 1^\circ$