Article

# IBN and Related Properties for Rings

Acta Mathematica Hungarica (Impact Factor: 0.35). 01/2002; 93(3):251-261. DOI: 10.1023/A:1015683326841

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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.06/2008: pages 281-293; - [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.Acta Mathematica Hungarica 03/2008; 119(1):95-125. · 0.35 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A ring R is called right strong stably finite (r.ssf) if for all n > 1, injective endomorphisms of R<sup>n</sup><sub>R</sub> are essential. If R is an r.ssf ring and eR is an idempotent of R such that eR is a retractable R -module, then eRe is an r.ssf ring. A direct product of rings is an r.ssf ring if and only if each factor is so. R.ssf condition is investigated for formal triangular matrix rings. In particular, if M is a finitely generated module over a commutative ring R such that for all n > 1, M <sup>( n )</sup> <sub>R</sub> is co-Hopfian, then egin{smallmatrix} End_R(M) & M\ 0 & R end{smallmatrix} is an r.ssf ring. If X is a right denominator set of regular elements of R , then R is an r.ssf ring if and only if RX <sup>–1</sup> is so.Acta Mathematica Universitatis Comenianae. 01/2009;

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