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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.