Abstract:
In this paper the author constructs a chain ring RR (i.e. a ring in which the right and left ideals are linearly ordered by inclusion) with the following properties: 1) RR is a prime ring; 2) the Jacobson radical J(R)J(R) of RR is a simple chain ring (without identity); 3) each element of J(R)J(R) is a right and left zero divisor. This example gives an answer to one of Brung's questions. In addition, the ring J(R)J(R) is totally singular, i.e. it coincides with its right (left) singular ideal.
The construction is based on a theorem that permits one to assign a chain ring to a right ordered group whose group ring can be imbedded in a division ring.
Bibliography: 9 titles.
\Bibitem{Dub83}
\by N.~I.~Dubrovin
\paper An example of a~chain prime ring with nilpotent elements
\jour Math. USSR-Sb.
\yr 1984
\vol 48
\issue 2
\pages 437--444
\mathnet{http://mi.mathnet.ru/eng/sm2140}
\crossref{https://doi.org/10.1070/SM1984v048n02ABEH002684}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=691988}
\zmath{https://zbmath.org/?q=an:0543.16003|0517.16002}
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https://doi.org/10.1070/SM1984v048n02ABEH002684
https://www.mathnet.ru/eng/sm/v162/i3/p441
This publication is cited in the following 22 articles:
Facchini A., Parolin C., “Rings Whose Proper Factors Are Right Perfect”, Colloq. Math., 122:2 (2011), 191–202
Chebotar M., Lee P.-H., Puczylowski E.R., “On Commutators and Nilpotent Elements in Simple Rings”, Bull. London Math. Soc., 42:Part 2 (2010), 191–194
H.H. Brungs, H. Marubayashi, E. Osmanagic, “Gauss extensions and total graded subrings for crossed product algebras”, Journal of Algebra, 316:1 (2007), 189
Miguel Ferrero, Ryszard Mazurek, Alveri Sant'Ana, “On right chain semigroups”, Journal of Algebra, 292:2 (2005), 574
Lee T., “Generalized Skew Derivations Characterized by Acting on Zero Products”, Pac. J. Math., 216:2 (2004), 293–301
Brungs, HH, “A classification and examples of rank one chain domains”, Transactions of the American Mathematical Society, 355:7 (2003), 2733
T. V. Dubrovina, N. I. Dubrovin, “On braid groups”, Sb. Math., 192:5 (2001), 693–703
N. I. Dubrovin, “Formal sums and power series over a group”, Sb. Math., 191:7 (2000), 955–971
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