# Unity of Ring is Unique

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## Contents

## Theorem

A ring can have no more than one unity.

## Proof

Let $\struct {R, +, \circ}$ be a ring.

If $\struct {R, \circ}$ has an identity, then it is a monoid.

From Identity of Monoid is Unique, it follows that such an identity is unique.

$\blacksquare$

## Also see

## Sources

- 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 1.3$: Some special classes of rings