A ring with several objects is a small additive category \({\mathcal C}\). In this setting, the role of the module category is played by the category \(Ab^{{\mathcal C}}\) of additive functors \({\mathcal C}\to Ab\). Extending results of the first author [Acta Math. Acad. Sci. Hung. 39, 185-193 (1982; Zbl 0495.16036)] various well-known results concerning Noetherian rings are carried over to this more general situation. In particular, \(\aleph_{\beta}\)-chain and \(\aleph_{\beta}\)-maximum conditions are introduced and shown to be equivalent for any object M of \(Ab^{{\mathcal C}}\). Here, the case \(\beta =-1\) corresponds to the usual conditions. Furthermore, examples of (ordinary) rings illustrating the concept for arbitrary \(\beta\) are given. With a suitable definition of left ideals, the usual characterizations of Noetherian rings generalize to \({\mathcal C}\). E.g. it is shown, that \({\mathcal C}\) satisfies the \(\aleph_{\beta}\)-chain condition for left ideals iff every \(\aleph_{\beta}\)-coproduct (a notion introduced in this paper) of injective objects of \(Ab^{{\mathcal C}}\) is injective. This coproduct is also used in connection with the decomposition of injective modules. In the last section, examples are given to show, that the injective hull and the \(\aleph_{\beta}\)- coproduct do not necessarily commute, not even over \({\mathbb{Z}}\).