  Yuri Matiyasevich's theorem states that the set of all Diophantine equations
which have a solution in non-negative integers is not recursive. Craig
Smory\'nski's theorem states that the set of all Diophantine equations which
have at most finitely many solutions in non-negative integers is not
recursively enumerable. Let R be a subring of Q with or without 1. By
H_{10}(R), we denote the problem of whether there exists an algorithm which for
any given Diophantine equation with integer coefficients, can decide whether or
not the equation has a solution in R. We prove that a positive solution to
H_{10}(R) implies that the set of all Diophantine equations with a finite
number of solutions in R is recursively enumerable. We show the converse
implication for every infinite set R \subseteq Q such that there exist
computable functions \tau_1,\tau_2:N \to Z which satisfy (\forall n \in N
\tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). This
implication for R=N guarantees that Smory\'nski's theorem follows from
Matiyasevich's theorem. Harvey Friedman conjectures that the set of all
polynomials of several variables with integer coefficients that have a rational
solution is not recursive. Harvey Friedman conjectures that the set of all
polynomials of several variables with integer coefficients that have only
finitely many rational solutions is not recursively enumerable. These
conjectures are equivalent by our results for R=Q.
