Chinese remainder theorem

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This article is about the statement of a simple but indispensable lemma in commutative algebra
View other indispensable lemmata


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