  Regarding polynomial functions on a subset $S$ of a non-commutative ring $R$,
that is, functions induced by polynomials in $R[x]$ (whose variable commutes
with the coefficients), we show connections between, on one hand, sets $S$ such
that the integer-valued polynomials on $S$ form a ring, and, on the other hand,
sets $S$ such that the set of polynomials in $R[x]$ that are zero on $S$ is an
ideal of $R[x]$.
