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No system of uncountable rank is purely simple


Author: Frank Okoh
Journal: Proc. Amer. Math. Soc. 79 (1980), 182-184
MSC: Primary 15A78; Secondary 15A21, 34D10
DOI: https://doi.org/10.1090/S0002-9939-1980-0565334-6
MathSciNet review: 565334
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Abstract: A pair of complex vector spaces (V, W) is a system if and only if there is a C-bilinear map $ {{\mathbf{C}}^2} \times V$ to W. The category of systems is equivalent to the category of modules over a certain subring of the ring of $ 3 \times 3$ matrices over the complex numbers, and so module-theoretic concepts make sense for systems. A system is purely simple if it has no proper pure subsystem. Recently it has been shown that for every positive integer n, there exists a purely simple system of rank n but no system of rank greater than the cardinality of the continuum is purely simple. In this paper it is shown that no system of rank greater than $ {\aleph _0}$ is purely simple. Necessary and sufficient conditions for a system of rank $ {\aleph _0}$ to be purely simple are also given.


References [Enhancements On Off] (What's this?)

  • [1] N. Aronszajn and U. Fixman, Algebraic spectral problems, Studia Math. 30 (1968), 273-338. MR 0240114 (39:1468)
  • [2] U. Fixman and F. Zorzitto, A purity criterion for pairs of linear transformations, Canad. J. Math. 26 (1974), 734-745. MR 0353054 (50:5540)
  • [3] F. Okoh, A bound on the rank of purely simple systems, Trans. Amer. Math. Soc. 232 (1977), 169-186. MR 0498625 (58:16711)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0565334-6
Keywords: System, purely simple, rank
Article copyright: © Copyright 1980 American Mathematical Society

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