Mirroring the construction of $\mathbb{Z}$ from $\mathbb{N}$, we can extend a commutative and additively cancellative semiring $A$ to its additive group of differences, $B$, and then define multiplication on $B$ by $$(a-b)(c-d):=ac+bd-(ad+bc),$$ forming a commutative ring with a natural embedding, $a\mapsto a-0$.
Under what conditions on $A$ is the constructed ring $B$ an integral domain? In particular, if $B$ is an integral domain, then $A$ is multiplicatively cancellative by its embedding, and conversely, as in the case for $\mathbb{N}$, if $A$ is multiplicatively cancellative and if each $a-b\in B$ corresponds to $c$ or $-c$, for $c\in A$, then $B$ is an integral domain. The latter implication seems too strong, so I would like to further refine the implications.
