The Chinese Remainder Theorem

The Chinese Remainder Theorem decomposes quotient rings along comaximal ideals, providing the algebraic foundation for modular arithmetic and more general ring decompositions.

Statement

Theorem4.5Chinese remainder theorem (general form)

Let $R$ be a commutative ring and $I_1, \ldots, I_n$ ideals that are pairwise comaximal: $I_j + I_k = R$ for all $j \neq k$ . Then:

$I_1 \cap I_2 \cap \cdots \cap I_n = I_1 I_2 \cdots I_n$ .
The natural map $\varphi: R \to R/I_1 \times R/I_2 \times \cdots \times R/I_n$ given by $r \mapsto (r + I_1, \ldots, r + I_n)$ is surjective with kernel $I_1 \cap \cdots \cap I_n$ . Hence:

$R/(I_1 \cap \cdots \cap I_n) \cong R/I_1 \times \cdots \times R/I_n.$

Proof

Case $n = 2$ : Since $I_1 + I_2 = R$ , write $1 = a + b$ with $a \in I_1$ , $b \in I_2$ .

Surjectivity: Given $(r_1 + I_1, r_2 + I_2)$ , the element $r = r_1 b + r_2 a$ satisfies $r \equiv r_1 \pmod{I_1}$ (since $a \in I_1$ , $b = 1 - a$ , so $r_1 b \equiv r_1$ ) and $r \equiv r_2 \pmod{I_2}$ (similarly).

Kernel: $\ker(\varphi) = I_1 \cap I_2$ . We show $I_1 \cap I_2 = I_1 I_2$ . Always $I_1 I_2 \subseteq I_1 \cap I_2$ . Conversely, if $c \in I_1 \cap I_2$ , then $c = c \cdot 1 = c(a + b) = ca + cb \in I_2 I_1 + I_1 I_2 = I_1 I_2$ .

Induction for $n > 2$ : We show $I_1$ and $I_2 \cap \cdots \cap I_n$ are comaximal. For each $k \geq 2$ , we have $a_k + b_k = 1$ with $a_k \in I_1$ , $b_k \in I_k$ . Then $\prod_{k=2}^{n} b_k \in I_2 \cdots I_n \subseteq I_2 \cap \cdots \cap I_n$ , and $1 - \prod b_k = 1 - \prod(1-a_k) \in I_1$ . So $I_1 + (I_2 \cap \cdots \cap I_n) = R$ , and we apply the $n = 2$ case. $\blacksquare$

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Applications

ExampleApplications of CRT

Classical CRT: $\mathbb{Z}/mn\mathbb{Z} \cong \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ when $\gcd(m,n) = 1$ . This gives the structure of $(\mathbb{Z}/n\mathbb{Z})^\times$ : $\varphi(n) = \prod \varphi(p_i^{e_i})$ .
Structure of finite abelian groups: Via the CRT applied to $\mathbb{Z}$ -modules, every finitely generated abelian group decomposes as $\bigoplus \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^r$ .
Polynomial interpolation: In $k[x]$ : the ideals $(x - a_1), \ldots, (x - a_n)$ are comaximal for distinct $a_i$ . CRT gives $k[x]/\prod(x-a_i) \cong k^n$ , recovering Lagrange interpolation.

RemarkNon-commutative and sheaf-theoretic generalizations

In algebraic geometry, the CRT globalizes to the sheaf condition: a sheaf is determined by its local sections, which must agree on overlaps. The CRT for rings is the affine case of this principle. In non-commutative algebra, analogous decomposition results hold for semisimple rings (Artin-Wedderburn theorem).