Category:Chinese Remainder Theorem (Commutative Algebra)

This category contains pages concerning Chinese Remainder Theorem (Commutative Algebra):

Let $A$ be a commutative and unitary ring.

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Let $I_1, \ldots, I_n$ for some $n \ge 1$ be ideals of $A$.

Then the ring homomorphism $\phi: A \to A / I_1 \times \cdots \times A / I_n$ defined as:

$\map \phi x = \tuple {x + I_1, \ldots, x + I_n}$

has the kernel $\ds I := \bigcap_{i \mathop = 1}^n I_i$, and is surjective if and only if the ideals are pairwise coprime, that is:

$\forall i \ne j: I_i + I_j = A$

Hence in that case, it induces an ring isomorphism:

$A / I \to A / I_1 \times \cdots \times A / I_n$

through the First Isomorphism Theorem.