Direct Sum of Rings?

1. Dec 26, 2009

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.

2. Dec 27, 2009

quasar987

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

3. Dec 27, 2009

Hurkyl

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