Basis with Different Cardinals in a R-Module

Vladimir Muka ()
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Vladimir Muka: Department of Mathematics, "Aleksander Moisiu" University

No 1401, Mathematical Modelling with Application in Economics from Technical University of Cluj Napoca, Department of Economics and Physics

Abstract: The study of modules "walks" in parallel with the study of vector spaces. Both of them are constructed on abelian groups. Many definitions and conclusions on vectorial spaces are equally valid for modules too. The only distinction between them is affiliation of scalars. In vector space the external multiplication is done by scalar taken from a field, while in modules the external multiplication is done by scalar taken from a ring. This seemingly small change brings substantial changes in the content of these algebraic structures. An important difference lies in the isomorphism of bases. Basis of a vector space are isomorphs between them, while in modules bases are not always isomorphs among them. In this article we will present an example that shows that there is at least a R-module containing bases with different cardinals. Our main objective in this article is to construct a special R-module that will contain subsets that are for it, the basis with different cardinals from each other. Furthermore, we will prove that R-module constructed for each natural number n has n bases with different cardinals. This example formulated as an exercise is in the book "Modules and the structure of the Rings", author Janothan S. Golan, Tom Head.

Keywords: module; vector space; cardinal's R module (search for similar items in EconPapers)
Pages: 7 pages
Date: 2014-06
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Published in Mathematical Modelling with Application in Economics, June 2014, pages 8-14

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