“Chinese remainder theorem”的概念、定义、翻译、参考文献-科学参考

Chinese remainder theorem

Mathematics

For coprime natural numbers m and n, ℤ_mn is isomorphic to ℤ_m×ℤ_n as a ring. More concretely, there is unique x in the range 0 ≤ x < mn that solves the simultaneous congruences x ≡ a (mod m) and x ≡ b (mod n). The isomorphism from ℤ_mn to ℤ_m×ℤ_n sends x mod mn to (x mod m, x mod n). If u,v are integers such that um + vn = 1 (see bézout’s lemma), then the inverse sends (a mod m, b mod n) to vna + umb mod mn.

More generally, if I and J are coprime ideals of a ring R then

$\frac{R}{I \cap J} ≅ \frac{R}{I} \times \frac{R}{J} .$

This can be proved by applying the first isomorphism theorem to the homomorphism r ↦ (r + I,r + J).

Computer

Let
$m_{1}, m_{2}, \dots, m_{r}$
be positive integers that are relatively prime to one another, and let their product be m:
$m = m_{1} m_{2} \dots m_{r}$
Let n, u₁, u₂,…, u_r be integers; then there is exactly one integer, u, that satisfies
$n \leq u < (m + n)$
and
$u \equiv u_{j} (modulo m_{j}) for 1 \leq j \leq r$

单词	Chinese remainder theorem
释义	Chinese remainder theorem Mathematics For coprime natural numbers m and n, ℤ_mn is isomorphic to ℤ_m×ℤ_n as a ring. More concretely, there is unique x in the range 0 ≤ x < mn that solves the simultaneous congruences x ≡ a (mod m) and x ≡ b (mod n). The isomorphism from ℤ_mn to ℤ_m×ℤ_n sends x mod mn to (x mod m, x mod n). If u,v are integers such that um + vn = 1 (see bézout’s lemma), then the inverse sends (a mod m, b mod n) to vna + umb mod mn. More generally, if I and J are coprime ideals of a ring R then $\frac{R}{I \cap J} ≅ \frac{R}{I} \times \frac{R}{J} .$ This can be proved by applying the first isomorphism theorem to the homomorphism r ↦ (r + I,r + J). Computer Let $m_{1}, m_{2}, \dots, m_{r}$ be positive integers that are relatively prime to one another, and let their product be m: $m = m_{1} m_{2} \dots m_{r}$ Let n, u₁, u₂,…, u_r be integers; then there is exactly one integer, u, that satisfies $n \leq u < (m + n)$ and $u \equiv u_{j} (modulo m_{j}) for 1 \leq j \leq r$
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