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On Roth’s pseudo equivalence over rings. (English) Zbl 1158.15009
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 $B=P'AQ'$.

15A24Matrix equations and identities
16E50Von Neumann regular rings and generalizations
Full Text: EuDML