No system of uncountable rank is purely simple

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.