nLab 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.

Related categories

• Ring

• Mod

References

• Stacks Project, 10.106 Epimorphisms of rings

category: category