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

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@boj54: The answer depends on much information which you haven't supplied, such as how the ring is "given", how you wish to "find" the ideals, etc. You'll need to supply such missing information before your question can be answered. – Gone Oct 1 '10 at 22:54
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@Bill Dubuque: I think boj54 read some theory of rings, and then the question sure is natural -- even though it is not possible. – AD. Oct 2 '10 at 6:30
@boj54: A similar problem would be How do you find the zeros of a polynomial? – AD. Oct 2 '10 at 6:34
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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? – Qiaochu Yuan Oct 2 '10 at 8:07

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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 $\mathbb{C}^2$.

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What do maximal ideals in $H^\infty$ look like? The corona theorem says that the open disk is dense in the maximal ideal space, but I for one could stand to better understand what a maximal ideal not in the open disk "looks like", aside from just knowing that very many of them exist. – Jonas Meyer Oct 2 '10 at 6:46
@Jonas Meyer: Let $A=H^infty(\mathbb{D})$, then first note that for each $\zeta\in\mathbb{D}$ the set $M_\zeta=\{f\in A: f(\zeta)=0\}$ is a maximal ideal space (because $(z-\zeta)\in M_\zeta$, and $g(z)=(f(z)-f(\zeta))/(z-\zeta)\in A$ if $f\in A$, but then $f(z) = f(\zeta) + (z-\zeta)g(z)$). However, there are also maximal ideal on the boundary of $\mathbb{D}$ but these are weak-* limits of point-evaluations which is the content of the corona theorem. (I'm not an expert on these but it is very interesting analysis - See the books Duren: H^p spaces, Garnett: Bounded analytic functions for more) – AD. Oct 2 '10 at 21:57
@AD.: Yes, that is what the corona theorem says. OK, so by saying that the problem of what "the maximal ideals [look] like...was solved", you just meant that it was proved that all of them are weak-∗ limit points of the disk. I wouldn't have used such language, but I wanted to clarify whether that was all you meant. I am interested in generalizations of the corona theorem, and I own copies of both of those excellent books you mentioned. (You meant "maximal ideal" where you wrote "maximal ideal space".) – Jonas Meyer Oct 4 '10 at 1:21
@Jonas Meyer: Ok, I am sorry about the miss-print. Are you into mathematical control theory? What kind of generalizations do you look at? – AD. Oct 4 '10 at 5:39
@AD.:(No need to apologize. This is one of those reasons it would be nice to have more than 5-minutes to edit.) I am not doing research in such generalizations, but I have seen some such work somewhat related to my research interests in the context of multiplier algebras of reproducing kernel Hilbert spaces. I really don't know much about those generalizations, but I'm interested and want to learn more. I also know very little control theory, but I find it interesting, too, from what I've seen. A gentle introduction is contained in Sasane's book books.google.com/books?id=1iSWPgAACAAJ&dq – Jonas Meyer Oct 5 '10 at 2:14
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