# Right Inverse for All is Left Inverse

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## Contents

## Theorem

Let $\struct {S, \circ}$ be a semigroup with a right identity $e_R$ such that:

- $\forall x \in S: \exists x_R: x \circ x_R = e_R$

That is, every element of $S$ has a right inverse with respect to the right identity.

Then $x_R \circ x = e_R$, that is, $x_R$ is also a left inverse with respect to the right identity.

## Proof

Let $y = x_R \circ x$. Then:

\(\displaystyle y \circ e_R\) | \(=\) | \(\displaystyle y \circ \paren {y \circ y_R}\) | Definition of Right Inverse Element | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {y \circ y} \circ y_R\) | as $\circ$ is associative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y \circ y_R\) | Product of Semigroup Element with Right Inverse is Idempotent: $y = x_R \circ x$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle e_R\) | Definition of Right Inverse Element |

So $x_R \circ x = e_R$, and $x_R$ behaves as a left inverse as well as a right inverse with respect to the right identity.

$\blacksquare$

## Also see

- Right Identity while exists Right Inverse for All is Identity
- Left Identity while exists Left Inverse for All is Identity

## Sources

- 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.4$: Lemmas $3$