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If you are given of a ring, how do you find its ideals?

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    I am curious what anyone can say about this question in general. That is, given an oracle describing addition and multiplication in a ring R, does there exist an algorithm which, given elements f_1, ... f_n, g of R, determines whether g is in the ideal generated by f_1, ... f_n?2010-10-02

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What Bill say in the comments is that there is no method to determine all ideals of a ring. Even if you specialize and look just for the maximal ideals in a given ring it might imply hard work!

A great example is to look at $H^\infty(\mathbb{D})$ which is the collection of all power series $f(z)=\sum_{n\ge0}a_nz^n$ such that $\sup_{|z|<1}|f(z)|<\infty.$ In other words, the space of all bounded analytic functions on the unit disc $\mathbb{D}$. It is fairly easy to see that $R=H^\infty(\mathbb{D})$ is a ring under point-wise operations. However, for long it was unknown how the maximal ideals ideals looks like. That problem was solved in 1962 by L. Carleson and is known as the Corona Theorem. I recommend reading this popular text regarding the Corona theorem.

In this day no one knows for sure how the the maximals looks like in $H^\infty(\mathbb{D}\times\mathbb{D})$, that is the space of bounded analytic functions in the bi-disc in $\mathbb{C}^2$.


Edit: The old link to the popular presentation of the Corona theorem has changed - using google with the following string

"corona theorem site:http://www.abelprize.no/" 

(without quotes) led me to the present link.

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    @rschwieb Thanks, I updated the link.2015-12-29