Finitely generated modules over pullback rings
The completion of the algebra \(k[x,y:xy=0]\) (\(k\) is a field) at its maximal ideal \(\langle x,y\rangle\) may be obtained as the pullback of two valuation domains mapping to a common factor field. \textit{L. Levy} [J. Algebra 93, 1-116 (1985; Zbl 0564.13010)] classified the finitely generated modules over such pullback (or, more generally, Dedekind-like) rings. In this paper the authors partly generalise Levy's results (to rings which are pullbacks of two hereditary noetherian homogeneously serial rings, finitely generated over their centres, mapping to a common semisimple artinian factor ring). Their approach is to analyse separately the maximal artinian submodule, \(A(M)\), and the quotient, \(M/A(M)\), of an indecomposable finitely generated module \(M\) (they show that these determine \(M\) among indecomposable modules). Their analysis goes via the associated graded ring of the pullback ring and, in particular, their analysis of the artinian modules uses the ``string and band'' classification [see, e.g., \textit{M. C. R. Butler} and \textit{C. M. Ringel}, Commun. Algebra 15, 145-179 (1987; Zbl 0612.16013)] of finite length modules over the first-mentioned algebra. A couple of typos: on p. 315 line 8, read ``\(JA\)'' not ``\(IA\)'' and, on the next line, insert ``homogeneously'' before the second occurrence of ``serial''.
hereditary Noetherian homogeneously serial rings
indecomposable finitely generated modules
scientific article; zbMATH DE number 934255