Category of commutative rings ($\mathbf{CRing}$)

The category of commutative rings, often written $\mathbf{CRing}$, has commutative rings with identity as objects and identity-preserving ring homomorphisms as morphisms. Composition and identity morphisms are the usual composition and identity maps. Including the zero ring makes it a terminal object; the integers form an initial object because a unique ring homomorphism sends $n$ to $n{1}_R$.

For example, a ring homomorphism $\mathbb{Z}[x]\to R$ is completely determined by the image of $x$, which can be any element of $R$. In algebraic geometry, applying the spectrum of a ring reverses arrows: a ring homomorphism $R\to S$ induces $\mathrm{Spec}S\to \mathrm{Spec}R$. Thus the category of affine schemes is equivalent to the opposite of the category of commutative rings.^[ref]