Editing Injective module (section)

=== Submodules, quotients, products, and sums, Bass-Papp Theorem===

Any [[product (category theory)|product]] of (even infinitely many) injective modules is injective; conversely, if a direct product of modules is injective, then each module is injective {{harv|Lam|1999|p=61}}. Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite [[direct sum of modules|direct sums]] of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is [[Artinian ring|Artinian]] [[semisimple ring|semisimple]] {{harv|Golan|Head|1991|p=152}}; every factor module of every injective module is injective if and only if the ring is [[hereditary ring|hereditary]], {{harv|Lam|1999|loc=Th. 3.22}}.

Bass-Papp Theorem states that every infinite direct sum of right (left) injective modules is injective if and only if the ring is right (left) [[Noetherian ring|Noetherian]], {{harv|Lam|1999|p=80-81|loc=Th 3.46}}.<ref>This is the [[Hyman Bass|Bass]]-Papp theorem, see {{harv|Papp|1959}} and {{harv|Chase|1960}}</ref>