Extending modules with injective or semisimple summands Original Research Article

We characterize when the direct sum of an extending module and an injective module is extending and when the direct sum of an extending module with the finite exchange property and a semisimple module is extending. We also characterize when the direct sum of a uniformextending module and a semisimple module is uniform-extending and, in consequence, we prove that, for a right Noetherian ring R, an extending right R-module M1 and a semisimple right R-module M2, the right R-module M1 circled plus M2 is extending if and only if m2 is M1/Soc(M1)-injective.