The usual ring operations on $\mathbb{Z}$ can be defined via polynomials in $\mathbb{Z}[a,b]$ (when viewing $a,b$ as variables):
Addition: $(a,b) \mapsto a+b \in \mathbb{Z}[a,b]$
Multiplication: $(a,b) \mapsto a \cdot b \in \mathbb{Z}[a,b]$
There are more ring structures on $\mathbb{Z}$ that can be described with polynomials, e.g. $(\mathbb{Z},P_A,P_M)$ with $P_A(a,b) = a+b-1$ and $P_M(a,b)=a+b-ab$. Here the neutral element w.r.t. $P_A$ is $1$ and those w.r.t. $P_M$ is $0$.
When we don't require the ring to have a neutral element w.r.t. multiplication, we could even take $P_A(a,b)=a+b$ and $P_M(a,b)=0$.
My question is: Can one characterize the pairs $(P_A,P_M)$ of polynomials, such that $(\mathbb{Z},P_A,P_M)$ is a ring? How does this change when we
- dont' require a neutral element w.r.t. $P_M$?
- want to get a commutative ring?
- want to get an integral domain?