The controller subgroup of one-sided ideals in completed group rings. (Q2895452)

When looking at right ideals \(I\) in a group ring \(k[G]\) over a field \(k\), it is often useful to know that the ideal \(I\) comes from a group ring \(k[H]\) of a subgroup \(H\subset G\), that is, \(I=I'\cdot k[G]\) for a right ideal \(I'\subset k[H]\). One sees easily that then one must have \(I'=I\cap k[H]\). Thus one is led to say that \(H\) ``controls'' \(I\) if \(I=(I\cap k[H])\cdot k[G]\), and to define \(I^\chi\), the so-called ``controller subgroup'', as the intersection of all subgroups \(H\) that control \(I\). Of course one must not jump to conclusions: it is not clear whether \(I^\chi\) itself controls \(I\). This happens exactly if \(I\) has a smallest \(H\) that controls it. As a certain step towards this statement, Passman proved that if two subgroups \(H_1\) and \(H_2\) control \(I\) then so does \(H_1\cap H_2\).NEWLINENEWLINE In this paper, these notions are adapted to profinite groups \(G\) and completed groups rings \(k[[G]]\). In the new definition of ``\(H\) controls \(I\)'', one restricts to closed subgroups \(H\), and takes the closure on the right hand side of the defining equation, i.e. \(I\) has to be topologically generated by an ideal \(I'\) of \(k[[H]]\), which then is again of necessity equal to \(I\cap k[[H]]\). In \S2 of the paper, Passman's result quoted above is transferred to the profinite group case, assuming that \(H_1\) and \(H_2\) are open. (In that case it is not necessary to take the closure, so we are back to the definition in the case of ordinary groups.) The controller subgroup \(I^\chi\) is now defined to be the intersection of all open controlling subgroups for \(I\). The obvious question (does \(I^\chi\) control \(I\)?) is left open in general, but answered affirmatively (Theorem A) for Iwasawa algebras, that is, for the case that \(G\) is (compact) \(p\)-adic analytic.NEWLINENEWLINE The general algebraic part \S2 makes use of so-called strongly \(G\)-graded algebras in a very nice way; even Hopf algebras put in a brief appearance. The proof of Theorem A is considerably more technical, beginning with a reduction to uniform groups and then using power series methods extensively. The author announces an application in a forthcoming paper: if \(G\) is, in addition, nilpotent, then every faithful (two-sided!) prime ideal is controlled by the centre of \(G\). This furnishes an exact analogue of a result of Zalesskii for groups without topology. All this is very clearly explained.NEWLINENEWLINEFor the entire collection see [Zbl 1232.16001].