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Regular rings, idempotents, and products of one-sided units. (English) Zbl 1100.16009

Let \(R\) be a one-sided unit-regular ring, i.e., for any \(a\) in \(R\) there exists a one-sided unit \(u\) in \(R\) such that \(a=aua\). In this paper the author proves that every element in \(R\) is the sum of an idempotent and a one-sided unit. Let \(S\) be a regular ring satisfying related comparability, i.e., \(S\) is a regular ring and \(S\) is the direct sum of \(A\) and \(B\), and is also the direct sum of \(C\) and \(D\), as \(S\)-modules, such that \(A\) isomorphic to \(C\), implies that there exists a central idempotent \(e\) in \(S\) such that \(Be\) is isomorphic to a submodule of \(De\) and \(D(1-e)\) is isomorphic to a submodule of \(B(1-e)\). The author proves also that every element in \(S\) is the sum of an idempotent and two one-sided units.

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16U60 Units, groups of units (associative rings and algebras)
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References:

[1] DOI: 10.1081/AGB-100002185 · Zbl 0992.16011 · doi:10.1081/AGB-100002185
[2] DOI: 10.1080/00927879708826068 · Zbl 0887.16001 · doi:10.1080/00927879708826068
[3] DOI: 10.1080/00927879908826691 · Zbl 0952.16010 · doi:10.1080/00927879908826691
[4] DOI: 10.1081/AGB-120037225 · Zbl 1073.16009 · doi:10.1081/AGB-120037225
[5] DOI: 10.1090/S0002-9904-1976-13980-8 · Zbl 0321.16016 · doi:10.1090/S0002-9904-1976-13980-8
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[7] DOI: 10.1090/S0002-9947-1977-0439876-2 · doi:10.1090/S0002-9947-1977-0439876-2
[8] DOI: 10.1081/AGB-100002409 · Zbl 0989.16015 · doi:10.1081/AGB-100002409
[9] DOI: 10.1090/S0002-9939-98-04397-4 · Zbl 0905.16009 · doi:10.1090/S0002-9939-98-04397-4
[10] DOI: 10.1017/S0017089504001727 · Zbl 1057.16007 · doi:10.1017/S0017089504001727
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