IBN and Related Properties for Rings

Acta Mathematica Hungarica (Impact Factor: 0.43). 06/2002; 93(3):251-261. DOI: 10.1023/A:1015683326841


We first tackle certain basic questions concerning the Invariant Basis Number (IBN) property and the universal stably finite
factor ring of a direct product of a family of rings. We then consider formal triangular matrix rings and obtain information
concerning IBN, rank condition, stable finiteness and strong rank condition of such rings. Finally it is shown that being
stably finite is a Morita invariant property.

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    ABSTRACT: We generalise the familiar notions of invariant basis number, rank condition, stable finiteness and strong rank condition from rings to modules. We study the inter relationship between these properties, identify various classes of modules possessing these properties and investigate the effect of many standard module theoretic operations on each one of these properties. We also tackle the important problem of preservation or non-preservation of these properties when we pass respectively to the module of polynomials, power series or inverse polynomials.
    No preview · Article · Mar 2008 · Acta Mathematica Hungarica
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    ABSTRACT: We carry out an extensive study of modules MR with the property that M/f(M) is singular for all injective endomorphisms f of M. Such modules called “quasi co-Hopfian”, generalize co-Hopfian modules. It is shown that a ring R is semisimple if and only if every quasi co-Hopfian R-module is co-Hopfian. Every module contains a unique largest fully invariant quasi co-Hopfian submodule. This submodule is determined for some modules including the semisimple ones. Over right nonsingular rings several equivalent conditions to being quasi co-Hopfian are given. Modules with all submodules quasi co-Hopfian are called “completely quasi co-Hopfian” (cqcH). Over right nonsingular rings and over certain right Noetherian rings, it is proved that every finite reduced rank module is cqcH. For a right nonsingular ring which is right semi-Artinian (resp. right FBN) the class of cqcH modules is the same as the class of finite reduced rank modules if and only if there are only finitely many isomorphism classes of nonsingular R-modules which are simple (resp. indecomposable injective).
    No preview · Article · May 2008 · Communications in Algebra
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    ABSTRACT: We call a ring R right Lazarus if any two maximal linearly independent subsets of a free right R-module have the same cardinality. We study these rings via weakly right semi-Steinitz rings. As an application, several classes of right Lazarus rings are given.
    Full-text · Chapter · Jun 2008
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