  We study systems of polynomial equations in infinite finitely generated
commutative associative rings with an identity element. For each such ring $R$
we obtain an interpretation by systems of equations of a ring of integers $O$
of a finite field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for
some prime $p$ and variable $t$. This implies that the Diophantine problem
(decidability of systems of polynomial equations) in $O$ is reducible to the
same problem in $R$. If, in particular, $R$ has positive characteristic or,
more generally, if $R$ has infinite rank, then we further obtain an
interpretation by systems of equations of the ring $\mathbb{F}_p[t]$ in $R$.
This implies that the Diophantine problem in $R$ is undecidable in this case.
In the remaining case where $R$ has finite rank and zero characteristic, we see
that $O$ is a ring of algebraic integers, and then the long-standing conjecture
that $\mathbb{Z}$ is always interpretable by systems of equations in a ring of
algebraic integers carries over to $R$. If true, it implies that the
Diophantine problem in $R$ is also undecidable. Thus, in this case the
Diophantine problem in every infinite finitely generated commutative unitary
ring is undecidable.
  The present is the first in a series of papers were we study the Diophantine
problem in different types of rings and algebras.
