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Direct Sum of Rings?

  1. Dec 26, 2009 #1
    What's the difference (if any) between a direct sum and a direct product of rings?

    For example, in Ireland and Rosen's number theory text, they mention that, in the context of rings, the Chinese Remainder Theorem implies that [tex]\mathbb{Z}/(m_1 \cdots m_n)\mathbb{Z}\cong\mathbb{Z}/m_1\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/m_n\mathbb{Z}[/tex]. The way I've been taught is that every [tex]\oplus[/tex] should be replaced with a [tex]\times[/tex] so that we are talking about direct products, not direct sums.
  2. jcsd
  3. Dec 27, 2009 #2


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    Yes, direct sum and direct product coincide for finitely many summands. You can check out the definition of both product on wikipedia.
  4. Dec 27, 2009 #3


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    He's not dealing with modules -- he's dealing with rings.
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