Let $A$ be a (commutative with unit) noetherian ring. If $I$ is an ideal of $A$, the $I$-adic completion of $A$ is by definition $$ \widehat{A} := \underset{\leftarrow}\lim A/I^n. $$ This operation is well known and has a lot of good properties, for example it is $I$-adically complete.

In an article I'm reading I found the sentence "completion at all prime ideals such that...". My question is: what is the completion with respect to a family of ideals?

My guess is the following: suppose we have finetely many ideals, say $I_1,\ldots,I_n$. Then the final result is obtained taking the completion wrt $I_1$, then the completion wrt (the ideal generated by) $I_2$ and so on. If we have infinitely many ideals we have to take a direct limit.

Is this construction discussed somewhere? Has it good properties? For example it's not totally clear to me that the order of the ideals doesn't affect the final result.

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