A note on termination of the Baer construction of the prime radical.
In this paper an algebra means an associative algebra (not necessarily with unity) over a fixed field \(F\). For a given algebra \(A\), \(W(A)\) denotes the sum of all nilpotent ideals of \(A\). For ordinal numbers \(\alpha\) the ideals \(W_\alpha(A)\) are defined inductively as follows: \(W_0(A)=0\). Suppose that \(\alpha>0\) and \(W_\beta(A)\) are defined for all \(\beta<\alpha\). Then, if \(\alpha\) is a limit ordinal number, then \(W_\alpha(A)=\bigcup_{\beta<\alpha}W_\beta(A)\). But, if \(\alpha\) is not a limit ordinal number, then \(W_\alpha(A)\) is the ideal of \(A\) containing \(W_{\alpha-1}(A)\) such that \(W_\alpha(A)/W_{\alpha-1}(A)=W(A/W_{\alpha-1}(A))\). It is well known that the above defined chain of ideals of \(A\) stabilizes at the prime radical \(\mathcal P(A)\) of \(A\). It is called the Baer construction of the prime radical. Amitsur proved that for every ordinal number \(\alpha>0\), there is an algebra \(A\) such that \(A=W_\alpha(A)\neq W_\beta(A)\) for every \(\beta<\alpha\). In such a situation we say that the Baer chain of \(A\) terminates at \(\alpha\). Bergman constructed an affine algebra of which the Baer chain terminates at 2. In this note the authors construct another and simpler example giving Amitsur's result that satisfies some extra conditions. They also construct for every countable non-limit ordinal number \(\alpha\), an affine algebra \(A\) such that the Baer chain of \(A\) terminates at \(\alpha\) which extends Bergman's result.
scientific article; zbMATH DE number 5814265