The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Eršov principle is proven for a theory of valued D-fields of residual characteristic zero.

The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Ersov principle is proven for a theory of valued D-fields of residual characteristic zero.

We describe classes of existentially closed ordered difference fields and rings. We show an Ax-Kochen type result for a class of valued ordered difference fields.

If T is a model complete theory with the strict order property, then the theory of the models of T with an automorphism has no model companion.