# nLab CRing

The category

### Context

#### Algebra

[[!include higher algebra - contents]]

#### Category theory

[[!include category theory - contents]]

# The category $CRing$

## Definition

$CRing$ is the category of commutative rings and ring homomorphisms.

A commutative ring is a commutative monoid object in Ab, so $CRing = CMon(Ab)$ is the category of commutative monoids in abelian groups.

The opposite category $CRing^{op}$ is the category of affine schemes.

## Properties

### Epi/Monomorphisms

Every surjective homomorphism of commutative rings is an epimorphism in $CRing$, but not every epimorphism is surjective.

A counterexample is the defining inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ of the ring of integers into the ring of rational numbers. This is an injective epimorphism of rings.

For more see for instance at Stacks Project, 10.106 Epimorphisms of rings.

### Cocartesian co-monoidal structure

###### Proposition

The coproduct in $CRing$ is given by the underlying tensor product of abelian groups, equipped with its canonically induced commutative ring structure.

By this general proposition discussed at category of commutative monoids.

###### Remark

Prop. means that tensor product of commutative rings exhibits cartesian monoidal category structure on the opposite category $CRing^{op}$.

## Generalizations

• The slice category of $CRing$ under a ring $R$ is the category $R$CAlg of commutative associative algebras over $R$.

• There is a “smooth” version of $CRing^{op}$: the category of smooth loci.

• There is a higher category theory version of $CRing$: the $(\infty,1)$-category of $E_\infty$-rings.

## References

category: category

Last revised on May 7, 2017 at 05:15:42. See the history of this page for a list of all contributions to it.