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On almost unit-regular rings. (English) Zbl 1267.16010

The author, in this paper, generalizes the notion of unit-regular ring to almost unit-regular ring. This notion is weaker than local ring but stronger than VNL ring. The author proves that an Abelian VNL ring is almost unit-regular. After deriving some preliminary results, in Section 2, he proves that a ring \(R\) is local if and only if its power series ring \(R[[X]]\) is almost unit-regular.
In Section 3, the author considers Morita context \((R,M,N,S)\) with zero pairings for bimodules \(_RM_S\) and \(_SN_R\). Here he proves that \(T\) is almost unit-regular if and only if the following conditions are satisfied: (i) Both \(R\) and \(S\) are almost unit-regular with one of them unit-regular; (ii) \((M,N)\) is either \(R\)-partial or \(S\)-partial; (iii) For any non-unit-regular \(r\) in \(R\), \((1-r)M=M\), \(N(1-r)=N\) and for any non-unit-regular \(s\) in \(S\), \(M(1-s)=M\), \((1-s)N=N\).
In Section 4, the author proves that a ring \(R\) is unit-regular if and only if \(M_2(R)\) is almost unit-regular. He derives some corollaries using his results on Morita contexts in Section 3.

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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References:

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