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 $r\ne s$. 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:

$I_1{I}_2\dots I_n=I_1\cap I_2\cap \dots \cap I_n$