No system of uncountable rank is purely simple
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- by Frank Okoh PDF
- Proc. Amer. Math. Soc. 79 (1980), 182-184 Request permission
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
- N. Aronszajn and U. Fixman, Algebraic spectral problems, Studia Math. 30 (1968), 273–338. MR 240114, DOI 10.4064/sm-30-3-273-338
- Uri Fixman and Frank A. Zorzitto, A purity criterion for pairs of linear transformations, Canadian J. Math. 26 (1974), 734–745. MR 353054, DOI 10.4153/CJM-1974-068-8
- Frank Okoh, A bound on the rank of purely simple systems, Trans. Amer. Math. Soc. 232 (1977), 169–186. MR 498625, DOI 10.1090/S0002-9947-1977-0498625-2
Additional Information
- © Copyright 1980 American Mathematical Society
- 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