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Pseudo-similarity and partial unit regularity. (English) Zbl 0544.15005
Let R be a ring and let A and B be square matrices of orders m and n respectively. A is said to be pseudo-similar to B if there is an \(m\times n\) matrix X and two possibly distinct \(n\times m\) matrices Y and Z such that \(YAX=B\), \(XAZ=A\), \(XYX=X\), \(XZX=X\). Recall that an element a in R is regular if there is a solution to the equation \(axa=a\) and unit regular (assume R has a unity) if there is a unit such that \(aua=a\). Finally R is called partially unit regular if every regular element is unit regular. The definition of pseudo-similarity can be stated for R as well.
The authors show that R is partially unit regular if and only if pseudo- similar elements of R are in fact similar. Applications are made to matrices over a principal ideal domain and to the reduction of regular matrices to a normal form.
Reviewer: G.P.Barker

MSC:
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A21 Canonical forms, reductions, classification
15A09 Theory of matrix inversion and generalized inverses
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References:
[1] T. P. Donovan: Certain matrix congruences mod \(p^n\). Ann. Mat. Pura Appl. IV, 65 (1977), 193-214. · Zbl 0374.15005
[2] F. J. Hall R. E. Hartwig I. J. Katz: A note on pseudo-similarity of matrices. J. of Industrial Math. Soc. 28 (1978), 25-36. · Zbl 0398.15009
[3] F. J. Hall I. J. Katz: On ranks of integral generalized inverses of integral matrices. Lin. and Mult. Alg. 7 (1979), 73-85. · Zbl 0398.15003
[4] R. E. Hartwig: How to partially order regular elements. Math. Japonica 25 (1980), 1-13. · Zbl 0442.06006
[5] R. E. Hartwig: Drazin inverses and canonical forms in \(M_n (Z/h)\). Lin. Alg. and Appl. 37 (1981), 205-233. · Zbl 0455.15003
[6] R. E. Hartwig F. J. Hall: Pseudo-similarity for matrices over a field. Proc. of AMS, 71 (1978), 6-10. · Zbl 0386.15013
[7] R. E. Hartwig J. Luh: A note on the group structure of unit regular ring elements. Pacific J. of Math. 71 (1977), 449-461. · Zbl 0355.16005
[8] R. E. Hartwig M. S. Putcha: Semi-similarity for matrices over a divison ring. Lin. Alg. and Appl. 39 (1981), 125-132. · Zbl 0467.15007
[9] M. Henriksen: On a class of regular rings that are elementary divisor rings. Archiv der Mathematik. 24 (1973), 133-141. · Zbl 0257.16015
[10] M. Newman: Integral matrices. Academic Press, New York) · Zbl 0254.15009
[11] E. D. Sontag: On generalized inverses of polynomial and other matrices. IEEE Trans. Automat. Contr. 25 (1980), 514-517. · Zbl 0447.15003
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