Abstract
A module M is said to be weakly injective if every finitely generated submodule of its injective hull E( M) is contained in a submodule X of E( M) isomorphic to M. We prove that a ring R satisfies the property that every cyclic right R-module has finite Goldie dimension if and only if every direct sum of (weakly) injective right R-modules is weakly injective. This is analog to the well-known characterization of right Noetherian rings as those for which direct sums of injective right modules are injective.
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