Unity of Ring is Unique

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

• Identity is Unique

Sources

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $3$: Some special classes of rings