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


15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A21 Canonical forms, reductions, classification
15A09 Theory of matrix inversion and generalized inverses
Full Text: EuDML


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