∑-extending modules Original Research Article

An R-module M is called ∑-extending if every coproduct of copies of M is extending, i.e. closed submodules are direct summands. Oshiro (1984) has shown that the ring R is ∑-extending as a left module if and only if the class of projective R->modules is closed under essential extensions. Using results from Garcia and Dung [5] on ∑-extending modules we generalize Oshiroʹs theorem to a wider class of modules. Under a weak projectivity condition we show that a module M is ∑-extending (equivalently, countably ∑-extending with ACC on M-annihilators) if and only if the class of direct summands of coproducts of copies of M is closed under M-generated essential extensions. Specializing to M = R our presentation offers alternative proofs to corresponding results for rings in (Oshiro, 1984) and (Vanaja, 1993). In addition we obtain that the ring R is left ∑-extending if and only if it is left countably ∑-extending and has ACC on left annihilators.