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A Dedekind algebra is an ordered pair (B, h), where B is a non-empty set and h is a similarity transformation on B. Among the Dedekind algebras is the sequence of the positive integers. From a contemporary perspective, Dedekind established that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. The purpose here is to show that this seemingly isolated result is a consequence of more general results in the model theory of second-order languages. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ℵ0 isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on ω called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type that occurs in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. The second-order theory of any countably infinite Dedekind algebra is categorical, and there are countably infinite Dedekind algebras whose second-order theories are not finitely axiomatizable. It is shown that there is a condition on configuration signatures necessary and sufficient for the second-order theory of a Dedekind algebra to be finitely axiomatizable. It follows that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. [ABSTRACT FROM AUTHOR]

We examine when summands of completely decomposable modules over a domain R are again completely decomposable. We show that this is the case if R is an h-local Prüfer domain. If R is 1-dimensional Noetherian, then the problem reduces locally if almost all localizations are integrally closed. If R is 1-dimensional Noetherian and local, then the integral closure of R must have at most two maximal ideals. [ABSTRACT FROM AUTHOR]

Let R be an integral domain with quotient field K and integral closure R'. Anderson and Zafrullah called R an "almost valuation domain" if for every nonzero x ∈ K, there is a positive integer n such that either xn ∈ R or x-n ∈ R. In this article, we introduce a new closely related class of integral domains. We define a prime ideal P of R to be a "pseudo-strongly prime ideal" if, whenever x, y ∈ K and xyP ⊆ P, then there is a positive integer m ≥ 1 such that either xm ∈ R or ymP ⊆ P. If each prime ideal of R is a pseudo-strongly prime ideal, then R is called a "pseudo-almost valuation domain" (PAVD). We show that the class of valuation domains, the class of pseudo-valuation domains, the class of almost valuation domains, and the class of almost pseudo-valuation domains are properly contained in the class of pseudo-almost valuation domains; also we show that the class of pseudo-almost valuation domains is properly contained in the class of quasilocal domains with linearly ordered prime ideals. Among the properties of PAVDs, we show that an integral domain R is a PAVD if and only if for every nonzero x ∈ K, there is a positive integer n ≥ 1 such that either xn ∈ R or ax-n ∈ R for every nonunit a ∈ R. We show that pseudo-almost valuation domains are precisely the pullbacks of almost valuation domains, we characterize pseudo-almost valuation domains of the form D + M, and we use this characterization to construct PAVDs that are not almost valuation domains. We show that if R is a Noetherian PAVD, then R has Krull dimension at most one and R' is a valuation domain; we show that every overring of a PAVD R is a PAVD iff R' is a valuation domain and every integral overring of R is a PAVD. [ABSTRACT FROM AUTHOR]

The structure of Ext is fundamental to understanding rings and their modules. For example, relative injectivity can be interpreted as a structural property of Ext as in when Ext[SUP1,SUBR](A, B) is torsion-free or zero. The purpose of this work is to examine Ext[SUP1,SUBR] (A,B under the assumption that A and B are torsion-free modules of finite rank, and R is a Dedekind domain. It is the abelian group theorist's mantra that their results extend to modules over Dedekind domains. This is almost always the case, however the structure of Ext is strongly influenced by the rank of the completion of R; unlike the case for the integers, the completion of a domain may have finite rank over the domain. Below, R will represent an integral domain with quotient field Q, and we will assume that R ≠ Q. In most case R will be assume to be Dedekind. All unadorned Ext, Hom, and ⊗ symbols are with respect to R. [ABSTRACT FROM AUTHOR]