# Is there a characterization of integral domains in terms of the homomorphisms out of them?

In the $\mathbf{Set}$-concrete category of commutative rings, we can define that an object $A$ is a field iff for every homomorphism $f : A \rightarrow B$, precisely one of the following holds.

1. $f$ is injective
2. $B$ is the trivial commutative ring.

(This only works if we assume that all our rings have a $1$ and that ring homomorphisms preserve $1$.)

Question. Is there a similar characterization of integral domains (viewed as commutative rings) in terms of the homomorphisms out of them?

• A stupid (i.e. not particularly useful) answer can be: There is an injection into a field (and use the characterization of fields described above)... – Pavel Čoupek Apr 13 '14 at 13:11
• One way of interpreting your characterization of a field is to say that a commutative ring $R$ is a field if and only if the lattice of equivalence classes of surjective homomorphisms with domain $R$ has exactly two elements. We can extend this approach to integral domains: A commutative ring $R$ is an integral domain if and only if the bottom element of the lattice of equivalence classes of surjective homomorphisms with domain $R$ is prime. – user45071 Jun 12 '14 at 7:50

A commutative ring $A$ is an integral domain $iff$ for all comm rings $B$ and all homomorphisms $f:B\rightarrow A$, $kerf$ is a prime ideal. This works by the first isomorphism and the fact that all subrings of integral domains are integral domains.
• That was the point of my comment above - since the lattice of equivalence classes of surjective homomorphisms with domain $R$ is isomorphic to the lattice of ideals of $R$, any property of rings which is statement about ideals can be characterized in terms of the corresponding fact about the surjection lattice. For instance the Noetherian and Artinian properties, being local, etc. all have interpretations as properties of the surjection lattice. – user45071 Jun 14 '14 at 5:29