# nLab CRing

### Context

#### Algebra

higher algebra

universal algebra

# 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

### 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. 1 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.

category: category

Revised on March 16, 2016 07:41:13 by Urs Schreiber (194.210.225.182)