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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
##### Keywords:
pseudo-similarity; unit regular rings; similarity; normal forms
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##### References:
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