ABSTRACT
The focus of this paper is the structure of ultraproducts of commutative rings, and in particular, ultraproducts of integral domains. Ultraproducts of certain classes of integral domains, such as orders in algebraic number fields, have been well-studied but often from a model-theoretic point of view. The model-theoretic view is a natural one to take given the importance of the ultraproduct construction in the theory of elementary models. In particular, the Keisler-Shelah Theorem states that two models have the property that any sentence in first-order logic satisfied by one is satisfied by the other if and only if these two models have isomorphic ultrapowers. In this same vein is the Ax-Kochen-Ershov Theorem, which implies that two henselian valuation domains have isomorphic ultrapowers if and only if their value groups and residue fields have isomorphic ultrapowers. Hence an understanding of arbitrary ultrapowers of a given class of commutative rings leads to a better understanding of the first order theory of the class of rings.
