TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The inclusion $\mathbb Z\hookrightarrow\mathbb Q$ is monic and epic but neither surjective nor an isomorphism in $\mathbf{Ring}$

Statement

In $\mathbf{Ring}$ , the canonical inclusion $i:\mathbb Z\hookrightarrow\mathbb Q$ is monic and epic, but its underlying function is not surjective and it is not an isomorphism.

Facts & Assumptions

Given: The canonical unital ring homomorphism $i:\mathbb Z\to\mathbb Q$ .

[L1]

The integers form a commutative ring (The integers form a commutative ring), the rationals form a field and hence a commutative ring (The rationals form a field, Every field is a commutative ring with $1 \ne 0$ ; it is an integral domain, and it is a commutative division ring), and The integers embed in the rationals identifies $i$ as an injective embedding.

[L2]

Morphisms of $\mathbf{Ring}$ are unit-preserving ring homomorphisms (Unital rings and unit-preserving ring homomorphisms form the large locally small category $\mathbf{Ring}$ ); monic, epic, and isomorphism mean cancellation and a two-sided inverse (Monomorphism and epimorphism by left and right cancellation, Isomorphism, groupoid, and connected category).

Proof

technique · direct

1.1

If $i\circ u=i\circ v$ , injectivity of $i$ gives $u=v$ pointwise, so $i$ is monic.

givenL1L2

2.1

If ring homomorphisms $f,g:\mathbb Q\to R$ agree after $i$ , then for $q=a/b$ with $b\ne0$ both send $q$ to the common image of $a$ times the inverse of the common image of $b$ ; hence $f(q)=g(q)$ for every $q$ , so $i$ is epic.

step 1.1L1L2

3.1

The rational $1/2$ is not an integer, so the underlying function of $i$ is not surjective; a categorical inverse would be an inverse function and would force surjectivity, so $i$ is not an isomorphism.

step 2.1L1L2∎

Depends on

Used by

Every morphism that is both monic and epic is an isomorphism False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 62 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)

The inclusion Z↪Q\mathbb Z\hookrightarrow\mathbb QZ↪Q is monic and epic but neither surjective nor an isomorphism in Ring\mathbf{Ring}Ring