nLab
CRing

The category CRingCRing

Definition

CRingCRing is the category of commutative rings and ring homomorphisms.

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

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

Properties

Cocartesian co-monoidal structure

Proposition

The coproduct in CRingCRing 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 opCRing^{op}.

Generalizations

category: category

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