PlanetMath: Chinese remainder theorem in terms of divisor theory

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Chinese remainder theorem in terms of divisor theory

(Theorem)

In a ring with a divisor theory, a congruence $\alpha \equiv \beta \pmod{\mathfrak{a}}$ with respect to a divisor module $\mathfrak{a}$ means that $\mathfrak{a} \mid \alpha\!-\!\beta$ .

Theorem. Let $\mathcal{O}$ be an integral domain having the divisor theory $\mathcal{O}^* \to \mathfrak{D}$ . For arbitrary pairwise coprime divisors $\mathfrak{a}_1,\,\ldots,\,\mathfrak{a}_s$ in $\mathfrak{D}$ and for arbitrary elements $\alpha_1,\,\ldots,\,\alpha_s$ of the domain $\mathcal{O}$ there exists an element $\xi$ in $\mathcal{O}$ such that

$\begin{align*}\begin{cases}\xi\, \equiv\, \alpha_1 \pmod{\mathfrak{a}_1}\\ \cdot... ...d \cdots\\ \xi\, \equiv\, \alpha_s \pmod{\mathfrak{a}_s} \end{cases}\end{align*}$

Proof. Let

$\displaystyle \mathfrak{b}_i \,:=\, \prod_{j \neq i}\mathfrak{a}_j \quad (i = 1,\,\ldots,\,s).$

Apparently, the divisors $\mathfrak{b}_1,\,\ldots,\,\mathfrak{b}_s$ are mutually coprime, whence there are in the ring $\mathcal{O}$ the elements $\beta_1,\,\ldots,\,\beta_s$ divisible by the divisors $\mathfrak{b}_1,\,\ldots,\,\mathfrak{b}_s$ , respectively, such that

$\displaystyle \beta_1+\ldots+\beta_s = 1.$

(1)

For every $i \neq j$ , the divisor $\mathfrak{a}_i$ divides $\mathfrak{b}_j$ and therefore also the element $\beta_j$ . Then the equation (1) implies that $\beta_i \equiv 1 \pmod{\mathfrak{a}_i}$ and thus the element

$\displaystyle \xi \,:=\, \alpha_1\beta_1+\ldots+\alpha_s\beta_s$

satisfies

$\displaystyle \xi \,\equiv\, \alpha_i\beta_i \,\equiv\, \alpha_i\! \pmod{\mathfrak{a}_i}$

for each $i = 1,\,\ldots,\,s$ . Q.E.D.

Bibliography

1: М. М. Постников: Введение в теорию алгебраических чисел. Издательство ``Наука''. Москва(1982).

"Chinese remainder theorem in terms of divisor theory" is owned by pahio.

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See Also: Chinese remainder theorem, Chinese remainder theorem

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Cross-references: implies, equation, divides, divisible, mutually coprime, pairwise coprime, integral domain, divisor, congruence, divisor theory, ring
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This is version 2 of Chinese remainder theorem in terms of divisor theory, born on 2008-04-28, modified 2008-04-29.
Object id is 10552, canonical name is ChineseRemainderTheoremInTermsOfDivisorTheory.
Accessed 251 times total.

Classification:

AMS MSC:	11A51 (Number theory :: Elementary number theory :: Factorization; primality)
	13A05 (Commutative rings and algebras :: General commutative ring theory :: Divisibility)

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