Let $\mathbb{N}_0 := \{0\} \cup \mathbb{N}$, where $\mathbb{N}=\{1,2,3,\dots\}$, equipped with its usual addition and multiplication. Define a relation $\sim$ on $\mathbb{N}_0 \times \mathbb{N}_0$ by declaring that, for all $a,b,c,d \in \mathbb{N}_0$,
are well-defined on $\mathbb{Z}_{\mathrm{fd}}$. With zero element $[0,0]$ and identity element $[1,0]$, these operations make $\mathbb{Z}_{\mathrm{fd}}$ a commutative ring with identity. The class $[a,b]$ represents the formal difference $a-b$.