Embeddable rings axiomatic class?

Question

In this question, a ring is defined to be with a unit distinct from the zero element, not necessarily a commutative ring though. Is the class of all such rings that can be embedded into fields an axiomatizable class? If so, what are the axioms?

Answer 1

Yes, such rings are exactly the integral domains, i.e. the rings which are commutative and in which $xy=0$ implies either $x=0$ or $y=0$. Any field is an integral domain (if $xy=0$ and $x\neq 0$, you can multiply by $x^{-1}$ to get $y=0$) and clearly any subring of an integral domain is an integral domain, so any ring that embeds in a field is an integral domain. Conversely, given an integral domain, it is a subring of its field of fractions.