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

Question

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?

Answer 1

I don't know how useful this is,

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.

The other way implication follows by taking the identity map and using zero ideal is prime iff the ring is an integral domain.

I don't know how to do this in terms of maps out of the integral domain.