Chinese remainder theorem

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

Suppose $I_{1}, I_{2}, \dots, I_{n}$ 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:

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

The injectivity of this map translates to the statement:

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