Divisible ℤ-modules

In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].Futa Yuichi - Japan Advanced Institute of Science and Technology Ishikawa, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205-214, 2012. doi:10.2478/v10037-012-0024-y.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Free ℤ-module. Formalized Mathematics, 20(4):275-280, 2012. doi:10.2478/v10037-012-0033-x.Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, and Yasunari Shidama. Torsion ℤ-module and torsion-free Z-module. Formalized Mathematics, 22(4):277-289, 2014. doi:10.2478/forma-2014-0028.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Torsion part of ℤ-module. Formalized Mathematics, 23(4):297-307, 2015. doi:10.1515/forma-2015-0024.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4), 1982.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective. The International Series in Engineering and Computer Science, 2002.Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990

Strong subgroup chains and the Baer-Specker group

Examples are given of non-elementary properties that are preserved under C-filtrations for various classes C of Abelian groups. The Baer-Specker group is never the union of a chain of proper subgroups with cotorsionfree quotients. Cotorsion-free groups form an abstract elementary class (AEC). The Kaplansky invariants of the Baer-Specker group are used to determine the AECs defined by the perps of the Baer-Specker quotient groups that are obtained by factoring the Baer-Specker group B of a ZFC extension by the Baer-Specker group A of the ground model, under various hypotheses, yielding information about its stability spectrum.Comment: 12 page

QUASI-IQC-INJECTIVITY

In this work, the notion of injectivity relative to a class of IQC sub modules (namely, IQC-injectivity) has been introduced and studied, which is a generalization quasi-injective module. This notion is closed under direct summands. Several properties and characterizations have been given.We provide a characterization of  semi simple  Artinian ring ,  SI-ring and Dedekind domain in terms of IQC-injective R-module

ON GENERALIZATION OF INJECTIVE MODULES

Here we introduce the concept of CK-N-injectivity as a gen-eralization of N-injectivity. We give a homomorphism diagram representation of such concept, as well as an equivalent condition in terms of module decompositions. The concept CK-N-jectivity is also dealt with, as a generalization of CK-N-injectivity. We introduce a generalization of N-injectivity, namely C-N-injectivity. Its generalization CI-N-injectivity (given in [8] as C-N-injectivity).In our study of C-N-injectivity, we discovered some mistake results (given in [1]as IC-Pseudo-injecyivity), and we dealt with their corrections. Finally we turn our attention to a more generalization of injective modules, namely the generalized extending modules (or module with (C1* )) and obtained some important results

Covering classes and 1-tilting cotorsion pairs over commutative rings

We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair provides for covers, that is when the class A is a covering class. We use Hrbekšfs bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R. Moreover, we use results of Bazzoni-Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if is the Gabriel topology associated to the 1-tilting cotorsion pair, and R is the ring of quotients with respect to, we show that if A is covering, then G is a perfect localisation (in Stenstromšfs sense [B. Stenstrom, Rings of Quotients, Springer, New York, 1975]) and the localisation R has projective dimension at most one as an R-module. Moreover, we show that is covering if and only if both the localisation RG and the quotient rings R/J are perfect rings for every J ∈. Rings satisfying the latter two conditions are called G-almost perfect

On the Reidemeister spectrum of an Abelian group

The Reidemeister number of an automorphism ϕ of an Abelian group G is calculated by determining the cardinality of the quotient group G/(ϕ − 1G)(G), and the Reidemeister spectrum of G is precisely the set of Reidemeister numbers of the automorphisms of G. In this work we determine the full spectrum of several types of group, paying particular attention to groups of torsion-free rank 1 and to direct sums and products. We show how to make use of strong realization results for Abelian groups to exhibit many groups where the Reidemeister number is infinite for all automorphisms; such groups then possess the so-called R∞-property.We also answer a query of Dekimpe and Gonçalves by exhibiting an Abelian 2-group which has the R∞-property

On FF-pure inversion of adjunction

We analyze adjunction and inversion of adjunction for the $F$-purity of divisor pairs in characteristic $p > 0$. In this vein, we give a complete answer for principal divisors under $\mathbb{Q}$-Gorenstein assumptions but without divisibility restrictions on the index. We also give a detailed analysis relating the $F$-purity of the pairs $(R,\Delta + D)$ and that of $(R_D, \text{Diff}_D(\Delta))$ motivated by Kawakita's log canonical inversion of adjunction via reduction to prime characteristic.Comment: 21 pages, comments welcom