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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?

Dannyu NDos
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    You might also like to convince yourself that the category of integral domains and injective ring homomorphisms not only has no terminal object but also no initial object. (This category is useful if you want to think about the field of fractions functor.) – Zhen Lin Feb 16 '23 at 01:37

1 Answers1

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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.

Dannyu NDos
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Mark Saving
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    Or, just consider $\mathbb{Z}[x]$, whose morphisms to $R$ are in bijection with elements of $R$. – Eric Wofsey Feb 16 '23 at 04:52
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    So, this even shows there is no weakly terminal object (i.e. an object for which there is at least one morphism from every other object to it, but not necessarily a unique such morphism). – Daniel Schepler Feb 16 '23 at 17:38