A Construction of Many Uncountable Rings Using Sfp Domains and Aronszajn Trees

The paper is in two parts. In Part I we describe a construction of a certain kind of subdirect product of a family of rings. We endow the index set of the family with the partial order structure of an SFP domain, as introduced by Plotkin, and provide a commuting system of homomorphisms between those rings whose indices are related in the ordering. We then take the subdirect product consisting of those elements of the direct product having finite support in the sense of this domain structure. In the special case where the homomorphisms are isomorphisms of a fixed ring S, our construction reduces to taking the Boolean power of 5 by a Boolean algebra canonically associated with the SFP domain. We examine the ideals of a ring obtainable in this way, showing for instance that each ideal is determined by its projections onto the factor rings. We give conditions on the underlying SFP domain that ensure that the ring is atomless. We examine the relationship between the L«,,0-theory of the ring and that of the SFP domain. In Part II we prove a 'non-structure theorem' by exhibiting 2 pairwise non-embeddable Loo(u-equivalent rings of cardinality K, with various higher-order properties. The construction needs only ZFC, and uses Aronszajn trees to build many different SFP domains with bases of cardinality K,.