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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.

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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 – 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

To shore up this question, the point is that infinite direct products are very different from infinite direct sums.

It is true that arbitrary products of injective modules are injective, and that a finite direct sum of injective modules is also a finite direct product and so is injective, but neither of these facts proves that an arbitrary (meaning possibly infinite) direct sum of injectives in injective.

To learn more about the proof that this is equivalent to Noetherianness of the ring, see this post: Projective and injective modules; direct sums and products.

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