Köthe’s famous open problem asks whether the nil radical \(\text{Nil}(R)\) of any ring \(R\) contains all nil left (or equivalently, right) ideals of \(R\) [G. Köthe, M. Z. 32, 161-186 (1930; JFM 56.0143.01)]. Recently it was shown that this is equivalent to ask whether \(A(R/A(R))=0\) for every ring \(R\) where, for a ring \(S\), \(A(S)\) denotes the ideal \(A(S)=\sum(L\mid L\) is a nil left ideal of \(S)\) [E. R. Puczyłowski, in Algebra and its applications. Proc. int. conf., Athens, OH 2005. Contemp. Math. 419, 269-283 (2006; Zbl 1133.16018)].
In pursuit of an answer to these problems, the authors define and study a chain of ideals \(\{A_\alpha(R)\}\), \(\alpha\) an ordinal number, for a ring \(R\) called the ‘Andrunakievich chain’ as follows: \(A_0(R)=0\). For \(\alpha>0\), suppose \(A_\beta(R)\) has been defined for all \(\beta<\alpha\). If \(\alpha\) is a limit ordinal, let \(A_\alpha(R)=\bigcup_{\beta<\alpha}A_\beta(R)\). If \(\alpha\) is not a limit ordinal, then \(A_\alpha(R)\) is the ideal of \(R\) containing \(A_{\alpha-1}(R)\) for which \(A(R/A_{\alpha-1}(R))=A_\alpha(R)/A_{\alpha-1}(R)\). One may then ask about the existence of an \(\alpha\) such that for every ring \(R\), the Andrunakievich chain of \(R\) stabilizes at an ordinal \(\leq\alpha\). The authors show that this question is also equivalent to Köthe’s problem.
In particular, it is shown that if Köthe’s problem has a negative solution, there is a field \(F\) such that for every ordinal number \(\alpha\) there is an \(F\)-algebra \(R_\alpha\) such that \(A_\alpha(R_\alpha)=R_\alpha\) and \(R_\alpha/A_\beta(R_\alpha)\) is not nil for all \(\beta<\alpha\) (i.e. \(A_\beta(R_\alpha)\neq R_\alpha\) for all \(\beta<\alpha\)). This construction is achieved by considering a certain matrix algebra over the field \(F\).