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Déterminants sur certains anneaux non commutatifs. (Determinants on some non commutative rings). (French) Zbl 0584.16012

The author asks for conditions on a ring A, under which a Dieudonné determinant can be defined; in particular he shows the following to be sufficient: Condition H. For any \((a,b)\in A^ 2\) there exist invertible elements \(\alpha\),\(\beta\),\(\gamma\) of A such that \(a=\alpha +\gamma\), \(b=\beta +\gamma^{-1}\). More precisely, over a ring A satisfying H, every square matrix with a right inverse is invertible and can be written as a product of elementary matrices and a diagonal matrix with one diagonal entry in the derived group of the group of units of A and the rest 1. The author also notes that H holds whenever A is a topological ring in which the group of units is a dense open subset.
Reviewer: P.M.Cohn

MSC:

16Kxx Division rings and semisimple Artin rings
15A15 Determinants, permanents, traces, other special matrix functions
16S50 Endomorphism rings; matrix rings
16U60 Units, groups of units (associative rings and algebras)
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