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Could people with knowledge in Commutative Rings elaborate on this sentence from the Wikipedia article (Ideals and Factor Rings subsection, first sentence):

The inner structure of a commutative ring is determined by considering its ideals, i.e. nonempty subsets that are closed under multiplication with arbitrary ring elements and addition

The sentence seems to imply that knowing what are the ideals in the ring, we learn alot about its structure. In particular, what do the ideals in a ring tell us about its structure..?

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    You can take this as a working definition of "inner structure." This will become clearer as you study more ring theory.2012-05-07
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    I think that Qiaochu is correct. Even if we explained why ideals are so important, even if we did it well, it won't have the same verve that it will when you finally "get it". I would say that you should sit down and start to get an intuition concerning this by figuring out why the only simple (i.e. no nontrivial ideals) [commutative] rings are the fields.2012-05-07
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    If there's a non-trivial ideal $I$ in a ring, then for any $x\in I$, $x$ does not have an inverse? Otherwise, $I$ would cover the whole ring.2012-05-07
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    As someone who has taught a graduate course in commutative algebra, let me say: it is not clear to me exactly what "inner structure" means here. On the other hand, it is certainly true that one learns an immense amount about the structure of a ring by studying its set of ideals (or prime ideals, or maximal ideals, or radical ideals, or Goldman ideals, or...). One needn't (and probably can't) explain all of the ways this occurs in advance, but it will show up naturally as one studies virtually any aspect of commutative ring theory.2012-05-07

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