Diophantine problems in rings and algebras: undecidability and reductions to rings of algebraic integers

Abstract

We study systems of equations in different families of rings and algebras. In each such structure R we interpret by systems of equations (e-interpret) a ring of integers O of a global field. The long standing conjecture that Z is always e-interpretable in O then carries over to R, and if true it implies that the Diophantine problem in R is undecidable. The conjecture is known to be true if O has positive characteristic, i.e. if O is not a ring of algebraic integers. As a corollary we describe families of structures where the Diophantine problem is undecidable, and in other cases we conjecture that it is so. In passing we obtain that the first order theory with constants of all the aforementioned structures R is undecidable.

On equations and first-order theory of one-relator monoids

“…The next two results are fundamental and they constitute the main reason we use interpretability in this paper. These are standard results whose proofs follow immediately from the Reduction Theorem 5.3.2 in [42] and Remark 3 after it (alternatively, see Lemma 2.7 of [37]).…”