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    ANDRUSZKIEWICZ, RYSZARD R. and SOBOLEWSKA, MAGDALENA 2013. ACCESSIBLE SUBRINGS AND KUROSH’S CHAINS OF ASSOCIATIVE RINGS. Journal of the Australian Mathematical Society, Vol. 95, Issue. 02, p. 145.


    ANDRUSZKIEWICZ, R. R. and SOBOLEWSKA, M. 2012. ON THE STABILISATION OF ONE-SIDED KUROSH’S CHAINS. Bulletin of the Australian Mathematical Society, Vol. 86, Issue. 03, p. 473.


    Niewieczerzal, Dorota 1999. A CONTRIBUTION OF ADAM SULIŃSKI TO RADICAL THEORY. Quaestiones Mathematicae, Vol. 22, Issue. 3, p. 449.


    BEIDAR, K.I. 1993. Theory of Radicals.


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Kurosh's chains of associative rings

  • R. R. Andruszkiewicz (a1) and E. R. Puczylowski (a2)
  • DOI: http://dx.doi.org/10.1017/S001708950000906X
  • Published online: 01 May 2009
Abstract

Let N be a homomorphically closed class of associative rings. Put N1 = Nl = N and, for ordinals a ≥ 2, define Nα (Nα) to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal (left ideal) in Nβ for some β<α. In this way we obtain a chain {Nα} ({Nα}), the union of which is equal to the lower radical class IN (lower left strong radical class IsN) determined by N. The chain {Nα} is called Kurosh's chain of N. Suliński, Anderson and Divinsky proved [7] that . Heinicke [3] constructed an example of N for which lNNk for k = 1, 2,. … In [1] Beidar solved the main problem in the area showing that for every natural number n ≥ 1 there exists a class N such that IN = Nn+l ≠ Nn. Some results concerning the termination of the chain {Nα} were obtained in [2,4]. In this paper we present some classes N with Nα = Nα for all α Using this and Beidar's example we prove that for every natural number n ≥ 1 there exists an N such that Nα = Nα for all α and NnNn+i = Nn+2. This in particular answers Question 6 of [4].

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2.N. Divinsky , J. Krempa and A. Suliński , Strong radical properties of alternative and associative rings, J. Algebra 17 (1971), 369381.

3.A. Heinicke , A note on lower radical constructions for associative rings, Canad. Math. Bull. 11 (1968), 2330.

4.E. R. Puczytowski , On questions concerning strong radicals of associative rings, Quaestiones Math. 10 (1987), 321338.

6.P. N. Stewart , On the lower radical construction, Ada Math. Acad. Sci. Hungar. 25 (1974), 3132.

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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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