Chinese remainder theorem

This article is about the statement of a simple but indispensable lemma in commutative algebra
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Statement

Suppose $I_{1},I_{2},\ldots ,I_{n}$ are ideals in a commutative unital ring $A$, with the property that any two of them are comaximal; in other words, $I_{r}+I_{s}=A$ for $r\neq s$. Then the natural map below is an isomorphism:

$A/(I_{1}I_{2}\ldots I_{n})\to A/I_{1}\times A/I_{2}\times \ldots A/I_{n}$

The injectivity of this map translates to the statement:

$I_{1}I_{2}\ldots I_{n}=I_{1}\cap I_{2}\cap \ldots \cap I_{n}$