Math 2.0

Commutative algebra Definition

Coprime ideals

Builds on Sum of ideals

Definition

Let RR be a commutative ring and let II and JJ be ideals of RR. The ideals II and JJ are coprime if I+J=RI + J = R, where I+JI + J is the sum of the ideals. Equivalently, there are a∈Ia \in I and b∈Jb \in J with a+b=1a + b = 1.

Remarks

In ℤ\mathbb{Z}, mℤ+nℤ=gcd⁡(m,n)ℤm\mathbb{Z} + n\mathbb{Z} = \gcd(m, n)\mathbb{Z}. So mℤm\mathbb{Z} and nℤn\mathbb{Z} are coprime if and only if gcd⁡(m,n)=1\gcd(m, n) = 1. If II and JJ are coprime, then I∩J=IJI \cap J = I J. Two distinct maximal ideals are always coprime.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionIdeal, definition, no evidence levelIdealdefinitionSum of ideals, concept, no evidence levelSum of idealsconceptCoprime ideals, definition, no evidence levelCoprime idealsdefinition
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  1. Group concept
  2. Ring definition
  3. Ideal definition
  4. Sum of ideals concept
  5. Coprime ideals definition

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2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Chinese_remainder_theorem, Generalization to arbitrary rings
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