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Zbl 1158.15009
Hartwig, R.E.; Patricio, Pedro
On Roth's pseudo equivalence over rings. (English)
[J] Electron. J. Linear Algebra 16, 111-124, electronic only (2007). ISSN 1081-3810/e

A von Neumann regular ring is stably finite if for every square matrices $A$ and $B$ with $AB=1$, one also has $BA=1$. Via an explicit calculation, it is shown that every block triangular matrix over such a ring is equivalent to its block diagonal. Combined with earlier results, this produces the main result (Theorem 5.1): a von Neumann regular ring is stably finite if and only if every block triangular matrix is equivalent to its block diagonal if and only if every block triangular matrix is pseudo-equivalent to its block diagonal. Two matrices $A$, $B$ of the same size are pseudo-equivalent if there are regular matrices $P$, $Q$ with pseudo-inverse $P'$ and $Q'$ such that $A=PBQ$ and $B=P'AQ'$.
[Gábor Braun (Budapest)]

MSC 2000:: *15A24 Matrix equations
16E50 Von Neumann regular rings and generalizations

Keywords: stably finite von Neumann regular rings; Roth equivalence

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