Exercise 4, § 2

Suppose that the additive group of $A$ contains an infinite subgroup $G$ whose non-zero elements are not divisors of zero in $A$. Show that the mapping $f \mapsto \tilde{f}$ of $A[X_1, \ldots, X_p]$ into the algebra of mappings of $A^p$ into $A$ is injective (observe that a polynomial of degree $n$ in one indeterminate cannot have more than $n$ distinct roots lying in $G$). This holds in particular when there is in $A$ an element $x$, not a divisor of $0$ and of infinite order in the additive group of $A$.