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On partially unit-regularity. (English) Zbl 1008.16009

Summary: We investigate partially unit-regularity. We show that \(R\) is partially unit-regular if and only if whenever \(ab\) and \(ba\) are strongly \(\pi\)-regular, there exists a \(u\in U(R)\) such that \((ab)^d=u(ba)^du^{-1}\). Furthermore, we show that if \(T\) is the ring of a Morita context \((A,B,M,N,\psi,\phi)\) with zero pairings, then \(T\) is partially unit-regular if and only if so are \(A\) and \(B\).

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D90 Module categories in associative algebras
16S50 Endomorphism rings; matrix rings
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