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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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
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.

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Additional Information

PII: S 0002-9939(1980)0565334-6
Keywords: System, purely simple, rank
Article copyright: © Copyright 1980 American Mathematical Society

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