Kurosh's chains of associative rings

Abstract

LetNbe a homomorphically closed class of associative rings. PutN1=Nl=Nand, for ordinalsa≥ 2, defineNα(Nα) to be the class of all associative ringsRsuch that every non-zero homomorphic image ofRcontains a non-zero ideal (left ideal) inNβfor some β<α. In this way we obtain a chain {Nα} ({Nα}), the union of which is equal to the lower radical classIN(lower left strong radical classIsN) determined byN. The chain {Nα} is calledKurosh's chainofN. Suliński, Anderson and Divinsky proved [7] that. Heinicke [3] constructed an example ofNfor whichlN≠Nkfork= 1, 2,. … In [1] Beidar solved the main problem in the area showing that for every natural numbern≥ 1 there exists a classNsuch thatIN=Nn+l ≠Nn. Some results concerning the termination of the chain {Nα} were obtained in [2,4]. In this paper we present some classesNwithNα=Nαfor all α Using this and Beidar's example we prove that for every natural numbern≥ 1 there exists anNsuch thatNα=Nαfor all α andNn≠Nn+i=Nn+2. This in particular answers Question 6 of [4].