Abstract:
We obtain criteria for the existence of a (left) unit in rings (arbitrary, Artinian, Noetherian, prime, and so on) that are based on the systematic study of properties of stable subsets of modules and their stabilizers that generalize the technique of idempotents. We study a class of quasiunitary rings that is a natural extension of classes of rings with unit and of von Neumann (weakly) regular rings, which inherits may properties of these classes. Some quasiunitary radicals of arbitrary rings are constructed, and the size of these radicals “measures the probability” of the existence of a unit.
Citation:
A. V. Khokhlov, “Stable subsets of modules and the existence of a unit in associative rings”, Mat. Zametki, 61:4 (1997), 596–611; Math. Notes, 61:4 (1997), 495–509
\Bibitem{Kho97}
\by A.~V.~Khokhlov
\paper Stable subsets of modules and the existence of a~unit in associative rings
\jour Mat. Zametki
\yr 1997
\vol 61
\issue 4
\pages 596--611
\mathnet{http://mi.mathnet.ru/mzm1538}
\crossref{https://doi.org/10.4213/mzm1538}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1620011}
\zmath{https://zbmath.org/?q=an:0949.16033}
\transl
\jour Math. Notes
\yr 1997
\vol 61
\issue 4
\pages 495--509
\crossref{https://doi.org/10.1007/BF02354994}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1997XR25700033}
Linking options:
https://www.mathnet.ru/eng/mzm1538
https://doi.org/10.4213/mzm1538
https://www.mathnet.ru/eng/mzm/v61/i4/p596
This publication is cited in the following 2 articles:
A. V. Khokhlov, “O suschestvovanii edinitsy v polukompaktnykh koltsakh i topologicheskikh koltsakh s usloviyami konechnosti”, Fundament. i prikl. matem., 8:1 (2002), 273–279
A. V. Khokhlov, “Stable Subsets and the Existence of a Unit in (Semi)-Prime Rings”, Math. Notes, 70:1 (2001), 123–131
Citation in format AMSBIB
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