Ring Epimorphism Preserves Unity
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Theorem
Let $A$ be a ring with unity $1$.
Let $B$ be a ring.
Let $f: A \to B$ be a ring epimorphism.
Then $\map f 1$ is a unity of $B$.
Proof
By definition, $f$ is a semigroup homomorphism between multiplicative semigroups.
A unity of a ring is by definition an identity element of its multiplicative semigroup.
Thus the result follows from Epimorphism Preserves Identity.
$\blacksquare$