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Equivalence of von Neumann regular and idempotent matrices. (English) Zbl 0801.15003
A matrix \(X\) is a 1-inverse for a matrix \(A\) if \(AXA = A\). \(X\) is a \(\{1,2\}\)-inverse of \(A\) if in addition \(XAX = X\). It is shwon that for matrices over certain types of rings that if \(A\) has a 1-inverse, then it also has a {1,2}-inverse.

15A09 Theory of matrix inversion and generalized inverses
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[1] F. J. Hall R. E. Hartwig I. J. Katz D. C. Newman: Pseudosimilarity and Partial Unit Regularity. Czechoslovak Mathematical Journal, 33 (108), Praha (1983).
[2] D. Handelman: Perspectivity and Cancellation in Regular Rings. Journal of Algebra 48, 1-16 (1977). · Zbl 0363.16009
[3] M. Henriksen: On a class of regular rings that are elementary divisor rings. Arch. Math. 24, 133-141 (1973). · Zbl 0257.16015
[4] D. Huylebrouck, J. Van Geel: Diagonalization of Idempotent Matrices. Journal of Algebra, Vol 105, no 1 January (1987). · Zbl 0606.15002
[5] N. Jacobson: Some remarks on one-sided inverses. Proc. Amer. Math. Soc. 1, 352-355 (1950). · Zbl 0037.15901
[6] A. V. Jategaonkar: Left Principal Ideal Rings. Lecture Notes in Mathematics 123, Springer, Berlin-New York (1970). · Zbl 0192.37901
[7] L. S. Levy, J. C. Robson: Matrices and Pairs of Modules. Journal of Algebra 29, 427-454 (1974). · Zbl 0282.16001
[8] R. Puystjens, J. Van Geel: On the Diagonalization of von Neumann Regular Matrices. Acta Universitatis Carolinae - Mathematica et Physica Vol. 26 (1985). · Zbl 0596.15012
[9] G. S. Rinehart: Note on the global dimension of certain rings. Proc. Amer. Math. Soc. 13, 341-346(1962). · Zbl 0104.26102
[10] A. Steger: Diagonability of Idempotent Matrices. Pac. Journal of Math. 19 nr. 3, 535-542 (1966). · Zbl 0147.29103
[11] R. B. Warfield, Jr.: Stable Equivalence of Matrices and Resolutions. Communications in Algebra, 6 (17, 1811-1828 (1978). · Zbl 0389.16008
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