Is there the terminal object in the category of integral domains?

Question

I wondered what the initial object and the terminal object are in the category of integral domains.

Simple argument: Since integral domains do not put an additional restriction on the definition of a ring homomorphism, integral domains should inherit the hom-set of rings (with unity).

That means the initial object in the category of rings, $\mathbb{Z}$, should be that of integral domains as well.

However, the terminal object in the category of rings, which is the zero ring $\mathbf{0}$, cannot be that of integral domains because, by definition, the zero ring is not an integral domain.

Am I overlooking something?

Answer 1

There is no terminal object in the category of integral domains.

For suppose $R$ is a terminal object. The existence of a ring homomorphism $\mathbb{F}_2 \to R$ means that $1 + 1 = 0$ in $R$. The existence of a ring homomorphism $\mathbb{F}_3 \to R$ means that $1 + 1 + 1 = 0$ in $R$. Therefore, $1 = 0$. This contradicts that $R$ is an integral domain.

If $0$ were an integral domain, then your argument would be valid. In general, if we have a full subcategory $C$ of $D$, then any limit diagram in $D$ is also a limit diagram in $C$, assuming all the objects in the diagram are in $C$. But as you noted, $0$ is not an integral domain. However, dually, $\mathbb{Z}$ is an integral domain and is therefore the initial object in the category of integral domains.