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Carlo Toffalori - One of the best experts on this subject based on the ideXlab platform.
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The torsionfree part of the Ziegler spectrum of orders over Dedekind Domains.
arXiv: Logic, 2018Co-Authors: Lorna Gregory, Sonia L'innocente, Carlo ToffaloriAbstract:We study the R-torsionfree part of the Ziegler spectrum of an order \Lambda over a Dedekind Domain R. We underline and comment on the role of lattices over \Lambda. We describe the torsionfree part of the spectrum when \Lambda is of finite lattice representation type.
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On pairs of free modules over a Dedekind Domain
Archive for Mathematical Logic, 2005Co-Authors: Saverio Cittadini, Carlo ToffaloriAbstract:The study of pairs of modules (over a Dedekind Domain) arises from two different perspectives, as a starting step in the analysis of tuples of submodules of a given module, or also as a particular case in the analysis of Abelian structures made by two modules and a morphism between them. We discuss how these two perspectives converge to pairs of modules, and we follow the latter one to obtain an alternative approach to the classification of pairs of torsionfree objects. Then we restrict our attention to pairs of free modules. Our main results are that the theory of pairs of free Abelian groups is co-recursively enumerable, and that a few remarkable extensions of this theory are decidable.
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the torsionfree part of the ziegler spectrum of rg when r is a Dedekind Domain and g is a finite group
Journal of Symbolic Logic, 2002Co-Authors: Annalisa Marcja, Mike Prest, Carlo ToffaloriAbstract:For every ring S with identity, the (right) Ziegler spectrum of S, Zg s , is the set of (isomorphism classes of) indecomposable pure injective (right) S-modules. The Ziegler topology equips Zg s with the structure of a topological space. A typical basic open set in this topology is of the form where φ and ψ are pp -formulas (with at most one free variable) in the first order language L s for S -modules; let [ φ / ψ ] denote the closed set Zg s - ( φ / ψ ). There is an alternative way to introduce the Ziegler topology on Zg s . For every choice of two f.p. (finitely presented) S-modules A, B and an S -module homomorphism f : A → B , consider the set ( f ) of the points N in Zg s such that some S -homomorphism h : A → N does not factor through f . Take ( f ) as a basic open set. The resulting topology on Zg s is, again, the Ziegler topology. The algebraic and model-theoretic relevance of the Ziegler topology is discussed in [Z], [P] and in many subsequent papers, including [P1], [P2] and [P3], for instance. Here we are interested in the Ziegler spectrum Zg RG of a group ring RG , where R is a Dedekind Domain of characteristic 0 (for example R could be the ring Z of integers) and G is a finite group. In particular we deal with the R -torsionfree points of Zg RG . The main motivation for this is the study of RG -lattices (i.e., finitely generated R -torsionfree RG -modules).
B. Sury - One of the best experts on this subject based on the ideXlab platform.
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A Dedekind Domain with Nontrivial Class Group
The American Mathematical Monthly, 2018Co-Authors: Vaibhav Pandey, Sagar Shrivastava, B. SuryAbstract:AbstractWe show that the ring of real-analytic functions on the unit circle is a Dedekind Domain with class number two.
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a Dedekind Domain with nontrivial class group
2016Co-Authors: Vaibhav Pandey, Sagar Shrivastava, B. SuryAbstract:Analytic properties of function spaces over the real and the complex fields are different in some ways. This reflects in algebraic properties which are different at times and similar in some other respects. For instance, the ring of real-valued continuous functions on a closed interval like $[0,1]$ behaves similarly to the corresponding ring of complex-valued functions; they depend only on the topology of $[0,1]$. The ring $\mathbf{R}[X,Y]/(X^2+Y^2-1)$ of real-valued polynomial functions on the unit circle is not a unique factorization Domain - witness the equation $$\cos^2(t) = (1+ \sin(t))(1- \sin(t)).$$ On the other hand, the ring $\mathbf{C}[X,Y]/(X^2+Y^2-1) \cong \mathbf{C}[X+iY, 1/(X+iY)]$ is a principal ideal Domain. Again, the rings of convergent power series (over either of these fields) with radius of convergence larger than some number $\rho$ is a Euclidean Domain (and hence, a principal ideal Domain) - this can be seen by using for a Euclidean "norm" function, the function which counts zeroes (with multiplicity) in the disc $|z| \leq \rho$. In this note, we consider the rings $C_{an}(S^1;\mathbf{R})$ of real-analytic functions on the unit circle $\mathit{S}^1$ which are real-valued and the corresponding ring $C_{an}(S^1; \mathbf{C})$ of analytic functions that are complex-valued. We will see that the latter is a principal ideal Domain while the former is a Dedekind Domain which is not a principal ideal Domain - the class group having order $2$.
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a ring for proposal of a nontrivial Dedekind Domain
arXiv: Rings and Algebras, 2016Co-Authors: Vaibhav Pandey, Sagar Shrivastava, B. SuryAbstract:Analytic properties of function spaces over the real and the complex fields are different in some ways. This reflects in algebraic properties which are different at times and similar in some other respects. For instance, the ring of real-valued continuous functions on a closed interval like $[0,1]$ behaves similarly to the corresponding ring of complex-valued functions; they depend only on the topology of $[0,1]$. The ring $\mathbf{R}[X,Y]/(X^2+Y^2-1)$ of real-valued polynomial functions on the unit circle is not a unique factorization Domain - witness the equation $$\cos^2(t) = (1+ \sin(t))(1- \sin(t)).$$ On the other hand, the ring $\mathbf{C}[X,Y]/(X^2+Y^2-1) \cong \mathbf{C}[X+iY, 1/(X+iY)]$ is a principal ideal Domain. Again, the rings of convergent power series (over either of these fields) with radius of convergence larger than some number $\rho$ is a Euclidean Domain (and hence, a principal ideal Domain) - this can be seen by using for a Euclidean "norm" function, the function which counts zeroes (with multiplicity) in the disc $|z| \leq \rho$. In this note, we consider the rings $C_{an}(S^1;\mathbf{R})$ of real-analytic functions on the unit circle $\mathit{S}^1$ which are real-valued and the corresponding ring $C_{an}(S^1; \mathbf{C})$ of analytic functions that are complex-valued. We will see that the latter is a principal ideal Domain while the former is a Dedekind Domain which is not a principal ideal Domain - the class group having order $2$.
S. Tariq Rizvi - One of the best experts on this subject based on the ideXlab platform.
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Baer module hulls of certain modules over a Dedekind Domain
Journal of Algebra and Its Applications, 2016Co-Authors: Jae Keol Park, S. Tariq RizviAbstract:The notion of a Baer ring, introduced by Kaplansky, has been extended to that of a Baer module using the endomorphism ring of a module in recent years. There do exist some results in the literature on Baer ring hulls of given rings. In contrast, the study of a Baer module hull of a given module remains wide open. In this paper we initiate this study. For a given module [Formula: see text], the Baer module hull, [Formula: see text], is the smallest Baer overmodule contained in a fixed injective hull [Formula: see text] of [Formula: see text]. For a certain class of modules [Formula: see text] over a commutative Noetherian Domain, we characterize all essential overmodules of [Formula: see text] which are Baer. As a consequence, it is shown that Baer module hulls exist for such modules over a Dedekind Domain. A precise description of such hulls is obtained. It is proved that a finitely generated module [Formula: see text] over a Dedekind Domain has a Baer module hull if and only if the torsion submodule [Formula: see text] of [Formula: see text] is semisimple. Further, in this case, the Baer module hull of [Formula: see text] is explicitly described. As applications, various properties and examples of Baer hulls are exhibited. It is shown that if [Formula: see text] are two modules with Baer hulls, [Formula: see text] may not have a Baer hull. On the other hand, the Baer module hull of the [Formula: see text]-module [Formula: see text] ([Formula: see text] a prime integer) is precisely given by [Formula: see text]. It is shown that infinitely generated modules over a Dedekind Domain may not have Baer module hulls.
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Baer module hulls of certain modules over a Dedekind Domain
Journal of Algebra and Its Applications, 2016Co-Authors: Jae Keol Park, S. Tariq RizviAbstract:The notion of a Baer ring, introduced by Kaplansky, has been extended to that of a Baer module using the endomorphism ring of a module in recent years. There do exist some results in the literature on Baer ring hulls of given rings. In contrast, the study of a Baer module hull of a given module remains wide open. In this paper we initiate this study. For a given module M, the Baer module hull, 𝔅(M), is the smallest Baer overmodule contained in a fixed injective hull E(M) of M. For a certain class of modules N over a commutative Noetherian Domain, we characterize all essential overmodules of N which are Baer. As a consequence, it is shown that Baer module hulls exist for such modules over a Dedekind Domain. A precise description of such hulls is obtained. It is proved that a finitely generated module N over a Dedekind Domain has a Baer module hull if and only if the torsion submodule t(N) of N is semisimple. Further, in this case, the Baer module hull of N is explicitly described. As applications, various ...
Simion Breaz - One of the best experts on this subject based on the ideXlab platform.
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Self-pure-generators over Dedekind Domains
Journal of Pure and Applied Algebra, 2019Co-Authors: Simion BreazAbstract:Abstract We prove that all pure submodules of a finite rank torsion-free module A over a Dedekind Domain are A -generated (i.e. A is a self-pure-generator) if and only if A has a rank 1 direct summand B such that type ( B ) is the inner type of A . This result is applied to describe the direct products of torsion-free groups of finite rank which are self-pure-generators.
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Self-pure-generators over Dedekind Domains
arXiv: Commutative Algebra, 2018Co-Authors: Simion BreazAbstract:We prove that all pure submodules of a finite rank torsion-free module $A$ over a Dedekind Domain are $A$-generated if and only if $A$ has a rank $1$ direct summand $B$ such that $\mathbf{type}(B)$ is the inner type of $A$.
P. F. Smith - One of the best experts on this subject based on the ideXlab platform.
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Direct products of simple modules over Dedekind Domains
Archiv der Mathematik, 2004Co-Authors: C. Santa-clara, P. F. SmithAbstract:Let R be a (commutative) Dedekind Domain and let the R -module M be a direct product of simple R -modules. Then any homomorphism from a closed submodule K of M to M can be lifted to M .
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Certain Chain Conditions in Modules over Dedekind Domains and Related Rings
Modules and Comodules, 1Co-Authors: Esperanza Sánchez Campos, P. F. SmithAbstract:Necessary and sufficient conditions are given for a module over a Dedekind Domain to satisfy the ascending chain condition on n-generated submodules for every positive integer n or on submodules with uniform dimension at most n for every positive integer n. These results are then extended to modules over commutative Noetherian Domains which need not be Dedekind.
