Direct sums and direct products of canonical pencils of matrices |
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Authors: | Frank Okoh |
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Institution: | Queen''s University Kingston, Ontario, Canada |
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Abstract: | A theorem of Kulikov characterizes the Kx]-modules which are direct sums of finite-dimensional (as a K-vector space) indecomposable modules, where Kx] is the polynomial ring over the field K. In this paper an analogous characterization is given for modules over the ring R, arising from pairs of linear transformations between a pair of complex vector spaces, (V, W). R is a certain subring of the ring of 3 × 3 complex matrices. The equivalence between the category of right R-modules and the category of systems enables one to work entirely in the category of systems. (A pair of complex vector spaces is a system if and only if there is a -bilinear map from to W). R-modules that are direct sums of finite-dimensional indecomposable subsystems are called pure-projective. The above characterization of pure-projective R-modules is used to prove that an R-module M is projective if and only if Ext(M, R) = 0. Direct products of finite-dimensional indecomposable R-modules are also studied, and a theorem pinpoints those that are pure-projective. An example of an R-module M that is not pure-projective, but with the property that every finite subset of M is contained in a pure-projective direct summand of M, is given. A by-product of this example is a class of matrices that generalizes the Vandermonde matrices. |
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