Chinese remainder theorem

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 $$r \ne s$$. Then the natural map below is an isomorphism:

$$A/(I_1I_2 \ldots I_n) \to A/I_1 \times A/I_2 \times \ldots A/I_n$$

The injectivity of this map translates to the statement:

$$I_1I_2\ldots I_n = I_1 \cap I_2 \cap \ldots \cap I_n$$