In the book Abstract Algebra of J.Antoine Grillet there is a theorem as follows:
A ring $R$ is left Noetherian if and only if every direct sum of injective left R-modules is injective
The Noetherian property is the core of the proof of this theorem in the book. However, I also know a proposition that says every direct product of injective modules is injective. In the finite case, the direct product and direct sum are the same, so the direct sum of injective modules is also injective.
So we do not need the Noetherian property anymore. Am I wrong?
Please explain for me. Thanks.