Relative purity over Noetherian rings

In this note we are going to show that if M is a left module over a left noetherian ring R of the infinite cardinality lambda >= vertical bar R vertical bar, then its injective hull E(M) is of the same size. Further, if M is an injective module with vertical bar M vertical bar >= (2(lambda))(+) and K <= M is its submodule such that vertical bar M/K vertical bar <= lambda, then K contains an injective submodule L with vertical bar M/L vertical bar <= 2(lambda). These results are applied to modules which are torsionfree with respect to a given hereditary torsion theory and generalize the results obtained by different methods in author's previous papers: [A note on pure subgroups, Contributions to General Algebra 12. Proceedings of the Vienna Conference, June 3-6, 1999, Verlag Johannes Heyn, Klagenfurt, 2000, pp. 105-107], [Pure subgroups, Math. Bohem. 126 (2001), 649-652]. (c) 2007 Mathematical Institute Slovak Academy of Sciences.