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 $CRing = CMon(Ab)$. 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 $CRing$; thus $CRing$ is cocartesian monoidal.
The opposite category $CRing^{op}$ is the category of affine schemes.
The slice category of $CRing$ under a ring $R$ is the categoy $R$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.