2
$\begingroup$

If $I_1,...,I_n$ are comaximal ideals in a commutative ring, then $I_1\cdots I_n=I_1\cap \cdots \cap I_n$. Does this extend to infinitely many comaximal ideals? The proof I have seen uses induction, so not sure if this does extend.

  • 15
    How do you define an infinite product of ideals?2011-12-20

1 Answers 1

2

Posting a CW Answer per @DylanMoreland's Comment:

The intersection of an arbitrary family of ideals makes sense, but the product of an infinite number of pairwise coprime ideals (even if the ring is commutative with unity) stands in need of definition. Of course one might choose to define the product in this case to agree with the intersection, but that kind of short-circuit is best avoided (unless it happens to serve an author's purpose).