Perhaps there is a more standard terminology, but let us say a subset $S$ of a commutative ring $R$ is a $\mathbb Z$-basis for $R$ if
- for every $a\in R$, there exist polynomial $p(\bar X)\in\mathbb Z[\bar X]$ and $\bar x\in S$ such that $a=p(\bar x)$; and
- for all polynomials $p(\bar X)\in\mathbb Z(\bar X)$ and all $\bar x\in S$, if $p(\bar x)=0$, then $p(\bar X)$ is the zero polynomial.
When does an ordered commutative ring have a $\mathbb Z$-basis? More generally, when can one find a $\mathbb Z$-basis which contains a given non-zero element for an ordered commutative ring?
The motivation is to have a cheap way of constructing ring automorphisms. What I can find do not seem to help. So references would be helpful too.
Let $A=\mathbb{Z}[X_s]$ be the polynomial ring with one variable for each element of $S$, and let $f:A\to R$ be the ring homomorphism sending $X_s$ to $s$.
Your first condition says that $f$ is surjective, and your second condition says that $f$ is injective. Therefore $f$ is an isomorphism, so $R$ must be a polynomial ring.