[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].


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