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The present paper is the result of a consideration of several related topics in the theory of linear associative algebras and the application of the results obtained to the theory of Riemann matrices. We first consider a linear algebra problem of great importance in its application to Riemann matrix theory, the question as to when a normal division algebra of order n2 over F is representable by an algebra of m-rowed square matrices with elements in F. It is shown that this is possible if anddoi:10.2307/1989468 fatcat:5kukltcszncp5pb3mtsr2nj35m