Injective module and Noetherian ring

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?

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If I understand you correctly, you've basically answered it yourself: a finite direct sum of injective modules over an arbitrary ring is injective, but that need not be the case for an infinite direct sum. – Martin Wanvik Apr 16 '12 at 18:00
Dear @MartinWanvik Are infinite direct sum and infinite direct product the same ? – Arsenaler Apr 16 '12 at 18:05
No, they're not (see en.wikipedia.org/wiki/Direct_sum_of_modules). – Martin Wanvik Apr 16 '12 at 18:08
@MartinWanvik : Thanks you very much ! – Arsenaler Apr 16 '12 at 18:13
Could you give me an example of an injective sum but not injective? – Miss Independent Jul 19 '12 at 14:01