From the introduction: ``Let \(G\) be a polycyclic-by-finite group, \(k\) a field, and \(V\) a right \(kG\)-module of finite \(k\)-dimension. This work was motivated by Musson's result [in I. M. Musson, Q. J. Math., Oxf. II. Ser. 31, 429--448 (1980; Zbl 0413.16012)], that if \(k\) has positive characteristic then the injective hull \(E(V)\) of \(V\) is Artinian. In this paper we study locally finite modules for \(kG\) where \(k\) has characteristic zero. A right module \(W\) for a \(k\)-algebra \(A\) is said to be locally finite (dimensional) if the \(k\)-dimension of \(wA\) is finite for each element \(w\) of \(W\). We obtain the characteristic-zero version of Musson's result and also show that the endomorphism ring \(\mathrm{End}_{kG}(E(V))\) is (right and left) Noetherian. Our other main results are that the completion \(widehat{kG}_I\) of \(kG\), with respect to an ideal \(I\) of finite codimension, is Noetherian and that if \(kG\) has enough clans (in the sense of \textit{B. J. Müller} [Can. J. Math. 28, 600--610 (1976; Zbl. 344.16004)] then \(G\) is nilpotent-by-finite. The starting point of our investigation was the observation that any locally finite right \(kG\)-module \(X\) may be naturally regarded as a left comodule for the Hopf algebra \(\mathcal F_0(G,k)\) of finitary functions on \(G\). Indeed in all our applications we exploit the remarkable fact that if \(X\) is an essential extension of a finite-dimensional \(kG\)-module and \(\mathrm{Hopf}(X)\) denotes the smallest sub-Hopf algebra of \(\mathcal F_0(G,k)\) over which it makes sense to regard \(X\) as a comodule then \(\mathrm{Hopf}(X)\) is a finitely generated \(k\)-algebra. Properties of a purely Hopf-theoretic nature which this calls into play are dealt with separately in the companion paper [\textit{S. Donkin}, J. Algebra 70, 394--419 (1981; Zbl 0464.16007)].''