Unity of Ring is Unique
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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): Chapter $1$: Rings - Definitions and Examples: $3$: Some special classes of rings