No system of uncountable rank is purely simple
Proc. Amer. Math. Soc. 79 (1980), 182-184
Primary 15A78; Secondary 15A21, 34D10
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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 to W. The category of systems is equivalent to the category of modules over a certain subring of the ring of 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 is purely simple. Necessary and sufficient conditions for a system of rank to be purely simple are also given.
Aronszajn and U.
Fixman, Algebraic spectral problems, Studia Math.
30 (1968), 273–338. MR 0240114
Fixman and Frank
A. Zorzitto, A purity criterion for pairs of linear
transformations, Canad. J. Math. 26 (1974),
734–745. MR 0353054
Okoh, A bound on the rank of purely simple
systems, Trans. Amer. Math. Soc. 232 (1977), 169–186. MR 0498625
(58 #16711), http://dx.doi.org/10.1090/S0002-9947-1977-0498625-2
- N. Aronszajn and U. Fixman, Algebraic spectral problems, Studia Math. 30 (1968), 273-338. MR 0240114 (39:1468)
- U. Fixman and F. Zorzitto, A purity criterion for pairs of linear transformations, Canad. J. Math. 26 (1974), 734-745. MR 0353054 (50:5540)
- 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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