Direct Sum vs Direct Product of Rings: Irish & Rosen

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SUMMARY

The discussion clarifies the distinction between direct sums and direct products of rings, specifically referencing the Chinese Remainder Theorem as presented in Ireland and Rosen's number theory text. It establishes that the isomorphism \(\mathbb{Z}/(m_1 \cdots m_n)\mathbb{Z} \cong \mathbb{Z}/m_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/m_n\mathbb{Z}\) illustrates a direct sum, while the speaker argues that this should be interpreted as a direct product. The consensus is that direct sums and direct products are equivalent only for finitely many summands, emphasizing the importance of context in their application.

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  • Understanding of ring theory and its definitions
  • Familiarity with the Chinese Remainder Theorem
  • Knowledge of isomorphisms in algebra
  • Basic concepts of modules and their distinctions from rings
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  • Explore the definitions and properties of direct sums and direct products of rings
  • Review the implications of isomorphisms in ring theory
  • Investigate the differences between modules and rings
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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 \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}. The way I've been taught is that every \oplus should be replaced with a \times so that we are talking about direct products, not direct sums.
 
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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.
 
He's not dealing with modules -- he's dealing with rings.
 

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