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Model Theory and Modules
, 1988
"... The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se ..."
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Cited by 73 (20 self)
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The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted ModR, the full subcategory of finitely presented modules will be denoted modR, the
Categorical quasivarieties via Morita equivalence
, 1994
"... Abstract. We give a new proof of the classification of ℵ0categorical quasivarieties by using Morita equivalence to reduce to term minimal quasivarieties. 1. ..."
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Abstract. We give a new proof of the classification of ℵ0categorical quasivarieties by using Morita equivalence to reduce to term minimal quasivarieties. 1.
A CONSTRUCTION OF MANY UNCOUNTABLE RINGS USING SFP DOMAINS AND ARONSZAJN TREES
, 1991
"... 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 ..."
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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«,,0theory of the ring and that of the SFP domain. In Part II we prove a 'nonstructure theorem ' by exhibiting 2 N  pairwise nonembeddable Loo(uequivalent rings of cardinality K, with various higherorder properties. The construction needs only ZFC, and uses Aronszajn trees to build many different SFP domains with bases of cardinality K,.
UNIVERSAL THEORIES CATEGORICAL IN POWER AND κGENERATED MODELS
"... Abstract. We investigate a notion called uniqueness in power κ that is akin to categoricity in power κ, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricitylike questions regarding ..."
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Abstract. We investigate a notion called uniqueness in power κ that is akin to categoricity in power κ, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricitylike questions regarding powers below the cardinality of a theory. We prove, for (uncountable) universal theories T, that if T is κunique for one uncountable κ, then it is κunique for every uncountable κ; in particular, it is categorical in powers greater than the cardinality of T. 404 revision:19940121 modified:19940129 It is well known that the notion of categoricity in power exhibits certain irregularities in “small ” cardinals, even when applied to such simple theories as universal Horn theories. For example, a countable universal Horn theory categorical in one uncountable power is necessarily categorical in all uncountable powers, by Morley’s theorem, but it need not be countably categorical. Tarski suggested that, for universal Horn theories T, this irregularity might be overcome by replacing the notion of categoricity in power by that of freeness in