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[parent] pairwise comaximal ideals property (Result)
Proposition 1   Let $ R$ be a commutative ring with unity. For every pairwise comaximal ideals $ I_1, I_2, ... , I_n$, the following holds:
$\displaystyle I_1 \cap I_2 \cap ... \cap I_n = I_1I_2 ... I_n.$ (1)

Proof. We prove by induction on $ n$. For $ n=2$, $ I_1+I_2 = R$ implies:
$\displaystyle I_1\cap I_2 = R(I_1\cap I_2) = (I_1 + I_2)(I_1 \cap I_2) \subseteq I_1I_2.$ (2)

The converse inclusion is trivial. Assume now that the equality holds for $ n \ge 2$: $ J:= I_1 \cap I_2 \cap ... \cap I_n = I_1I_2 ... I_n$. Since $ I_{n+1}+I_j = R$, for every $ j \neq {n+1}$, there exist the elements $ a_j\in I_j$ and $ b_j\in I_{n+1}$ such that $ a_j + b_j =1$. The product $ c:= \prod_{j=1}^n a_j = \prod_{j=1}^n(1-b_j)\in 1 + I_{n+1}$. Also $ c\in J$, then $ 1\in J+I_{n+1}$ or $ J+I_{n+1} =R$.
Applying the case $ 2$, the induction step is satisfied:
$\displaystyle I_1I_2 ... I_{n+1} = J I_{n+1} = J\cap I_{n+1} = I_1\cap I_2 \cap ... \cap I_n \cap I_{n+1}.$ (3)

$ \qedsymbol$



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Cross-references: product, equality, inclusion, converse, implies, induction, comaximal ideals, unity, commutative ring

This is version 6 of pairwise comaximal ideals property, born on 2007-04-04, modified 2007-04-05.
Object id is 9149, canonical name is DerivationOfComaximalIdeals.
Accessed 335 times total.

Classification:
AMS MSC: 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
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