Geodesic
Von Neumann's Manuscript on Inductive Limits of Regular Rings
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 477-483

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It is now known (3) that if is a regular rank ring, then the rank function can be extended to the matrix ring in such a way that R(a) = R(a ꕕ n) ; here, a is an arbitrary element of is the n × n diagonal matrix with a for each entry on the diagonal, and R denotes rank in and also in . It is also known (2) that every regular rank ring has a rankmetric completion which is again a regular rank ring. 
Halperin, Israel. Von Neumann's Manuscript on Inductive Limits of Regular Rings. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 477-483. doi: 10.4153/CJM-1968-045-0
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[1] 1. Dawkins, Brian P. and Halperin, Israel, The isomorphism of certain continuous rings, Can. J. Math., 18 (1966), 1333–1344. Google Scholar

[2] 2. Halperin, Israel, Regular rank rings, Can. J. Math., 17 (1965), 709–719. Google Scholar

[3] 3. Halperin, Israel, Extension of the rank function, Studia Math. 27 (1966), 325–335. Google Scholar

[4] 4. von Neumann, John, Continuous geometry (Princeton, 1960). Google Scholar

[5] 5. von Neumann, John, Independence of F-from the sequencey, unpublished manuscript written in 1936–37 review by I. Halperin in Vol. IV of the Collected Works of John von Neumann Pergamon, (1962). Google Scholar

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