The influence of ideal structure on representation theory
Let R be a Noetherian ring. The existence of a uniform right R-module M that is the non-split extension of a uniform submodule U with \(Ann(U)=Ass(U)=P\) and \(U=Ann_ M(P)\) by a uniform module V with \(Ann(V)=Ass(V)=Q\), allows considerable insight into the prime ideal structure of R. Thus, \textit{A. V. Jategaonkar}'s ``Main Lemma'' [Localization in Noetherian Rings (Lond. Math. Soc. Lect. Note Ser. 98, 1986; Zbl 0589.16014)] shows that either Q is properly contained in P, or else there is a link from Q to P. In the first three sections of the paper, the authors offer a number of sufficient conditions on P, Q, U, and V for the existence of such an extension. As an application, they answer negatively a long outstanding question whether the annihilator of a (Krull) critical right R-module always has to be prime, as is the case when the Noetherian ring R is ideal invariant. In the last two sections, the impact of the link structure of the prime spectrum on the layered structure of an injective indecomposable module is studied, and the layers are described in analogy with the pioneer work for commutative Noetherian rings done by \textit{E. Matlis} [Pac. J. Math. 8, 511-528 (1958; Zbl 0084.266)]. This investigation is naturally restricted to rings satisfying the strong second layer condition, because the link structure is best understood for these rings, a class which is vast enough, as it comprises such rings as group rings of polycyclic-by-finite groups, enveloping algebras of finite-dimensional solvable Lie algebras over algebraically closed fields, Ore and skew polynomial extensions of commutative Noetherian rings, and FBN rings. The best results are obtained for polynormal rings and for group rings of polycyclic-by-finite groups with commutative Noetherian coefficient rings. The additional insight is possible because in these cases the link graphs are locally finite.
non-split extension of a uniform submodule
group rings of polycyclic-by-finite groups
enveloping algebras of finite-dimensional solvable Lie algebras
scientific article; zbMATH DE number 4063294
scientific article