Glasgow Mathematical Journal

Research Article

Kurosh's chains of associative rings

R. R. Andruszkiewicza1 and E. R. Puczylowskia2

a1 Institute of Mathematics University of Warsaw, Bialystok Division Akademicka 2, 15-267 Bialystok Poland

a2 Institute of Mathematics University of Warsaw Pkin, 00-901 Warsaw Poland

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 S001708950000906X_inline1. 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].

(Received October 01 1988)