nLab
CRing

Definition

$\mathrm{CRing}$ is the category of commutative rings and ring homomorphisms.

A commutative ring is a commutative monoid object in Ab, so $\mathrm{CRing}=\mathrm{CMon}(\mathrm{Ab})$. As for commutative monoid objects in any symmetric monoidal category, the tensor product of commutative rings is again a commutative ring, and is the coproduct in $\mathrm{CRing}$; thus $\mathrm{CRing}$ is cocartesian monoidal.

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

Generalizations

• The slice category of $\mathrm{CRing}$ under a ring $R$ is the categoy $R$CAlg of commutative associative algebras over $R$.

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

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

category: category