Math 2.0

Commutative algebra Basic notions Theorem

Chinese remainder theorem for rings

Builds on Coprime idealsQuotient ringProduct of ringsRing isomorphism

Statement

Let RR be a commutative ring and let I1,…,InI_1, \dots, I_n be ideals of RR, with n≥1n \ge 1. Suppose that IjI_j and IkI_k are coprime for all j≠kj \neq k. Then the map R/(I1∩⋯∩In)→R/I1×⋯×R/In,a+(I1∩⋯∩In)↦(a+I1,…,a+In)R/(I_1 \cap \dots \cap I_n) \to R/I_1 \times \dots \times R/I_n, \qquad a + (I_1 \cap \dots \cap I_n) \mapsto (a + I_1, \dots, a + I_n) is a ring isomorphism.

Remarks

Under the same hypothesis, I1∩⋯∩In=I1I2⋯InI_1 \cap \dots \cap I_n = I_1 I_2 \cdots I_n. For R=ℤR = \mathbb{Z} and pairwise coprime integers m1,…,mnm_1, \dots, m_n, the theorem gives ℤ/(m1⋯mn)ℤ≅ℤ/m1ℤ×⋯×ℤ/mnℤ\mathbb{Z}/(m_1 \cdots m_n)\mathbb{Z} \cong \mathbb{Z}/m_1\mathbb{Z} \times \dots \times \mathbb{Z}/m_n\mathbb{Z}. For example, ℤ/6ℤ≅ℤ/2ℤ×ℤ/3ℤ\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionIdeal, definition, no evidence levelIdealdefinitionProduct of rings, definition, no evidence levelProduct of ringsdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionSum of ideals, concept, no evidence levelSum of idealsconceptTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionCoprime ideals, definition, no evidence levelCoprime idealsdefinitionQuotient ring, definition, no evidence levelQuotient ringdefinitionChinese remainder theorem for rings, statement, no evidence levelChinese remainder the…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Ideal definition
  4. Product of rings definition
  5. Ring homomorphism definition
  6. Sum of ideals concept
  7. Two-sided ideal definition
  8. Ring isomorphism definition
  9. Coprime ideals definition
  10. Quotient ring definition
  11. Chinese remainder theorem for rings statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Chinese_remainder_theorem, Generalization to arbitrary rings
About this page
Tag0W97J8QED7
URIm2://0W97J8QED7
Kindstatement, theorem
WritingW0 stub
MSC13A15
StatusThis copy of the site has no build data. Run m2 status 0W97J8QED7.
UsesCoprime ideals, Quotient ring, Product of rings, Ring isomorphism

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