symmetric monoidal (∞,1)-category of spectra
$CRing$ 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.
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.
Prop. 1 means that tensor product of commutative rings exhibits cartesian monoidal category structure on the opposite category $CRing^{op}$.
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.