IBN and Related Properties for Rings

ArticleinActa Mathematica Hungarica 93(3):251-261 · June 2002with8 Reads
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.
    • "Another terminology which has been used for the UGN property is the " rank condition " (see, e.g., [13] and [14, Section 1C] ). We note the following easily verified equivalent characterizations of the UGN property. "
    [Show abstract] [Hide abstract] ABSTRACT: We present a result of P. Ara which establishes that the Unbounded Generating Number property is a Morita invariant for unital rings. Using this, we give necessary and sufficient conditions on a graph $E$ so that the Leavitt path algebra associated to $E$ has UGN. We conclude by identifying the graphs for which the Leavitt path algebra is (equivalently) directly finite; stably finite; Hermite; and has cancellation of projectives.
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  • [Show abstract] [Hide abstract] 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.
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  • [Show abstract] [Hide abstract] 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).
    Article · May 2008
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