[FOM] "Diophantine Complete" rings
Fred Richman
richman at fau.edu
Sun Jul 6 10:12:09 EDT 2003
Martin Davis wrote:
> Harvey Friedman has proposed the study of Diophantine
> Complete (DC) rings, that is rings with the property that
> whenever a polynomial with integer coefficients has a zero
> in the ring, it also has an zero in the integers.
I wonder why this is called "Diophantine Complete". It's
really a relationship between a ring and a subring (in this
case the subring is the ring of integers Z) in which the
subring has the completeness property: the subring is
something like algebraically closed relative to the ring.
The analogous condition for abelian groups is described by
saying that the subgroup is pure in the group.
> I note that any ring which can be homomorphically mapped
> into the integers is obviously DC.
> Question: Are there any others?
For b a transcendental p-adic integer, the ring R of p-adic
integers r such that (p^n)r is in Z[b] for some positive
integer n is DC. In fact, if r satisfies a nontrivial
polynomial over Z, then r is in Z. Moreover, R is
indecomposable as an abelian group, so it can't admit a
homomorphism onto Z.
> Obvious examples of rings for which such homomorphisms exist
> are the rings of polynomials in some set of indeterminates.
> Are there any others?
Any subring of such a ring, like Z[x^2, x^3], and lots of
quotient rings, like Z[x,y]/(x^2 + y^2 - 1].
--Fred
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