Direct sum vs direct product of rings

[altcmdesc]

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.

 

[quasar987]

Yes, direct sum and direct product coincide for finitely many summands. You can check out the definition of both product on wikipedia.

 

[Hurkyl]

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