# What do ideals of a ring say about its inner structure?

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. – Qiaochu Yuan May 7 '12 at 2:37
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. – Alex Youcis May 7 '12 at 2:46
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. – wu7 May 7 '12 at 3:48
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. – Pete L. Clark May 7 '12 at 4:21

One needs to be careful about how much to expect from this claim. Much can be learned from the ideals, but not everything. An obvious example would be to think of fields, and how diverse and interesting they are, but their ideal structures are all identically boring. This can still occur for rings with proper ideals, too. When there are nontrivial ideals, and you can track their products things become more interesting, but still, not everything is there.

A better try is to study not only the ideals, but the entire category of modules for the ring. It is much richer and complete in information about the ring. This is because the ideals (and right and left ideals, if we want to talk about noncommutative rings) contain information about the ring (and module) homomorphisms beginning in $R$.

(Digression: In order to study ideals of commutative rings, Ward and Dilworth abstracted out to the concept of "multiplicative lattices". They got a certain level of structure out of it.)

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Could you give an example of a property of a ring not following from its ideals but only from its modules? – wu7 May 10 '12 at 3:29
All right ideals projective= right hereditary. All right modules projective= semisimple. If you are searching for a module-characterized ring property with absolutely no ideal characterization, nothing jumps to mind for me right away. – rschwieb May 10 '12 at 11:05

The statement is too vague to determine the author's intent. Perhaps the author intends to contrast the study of structure of rings using internal structure such as ideals vs. external structure such as general modules over the ring. Regarding such, you might find of interest the following nice introduction to J.P. Lafon: Ideals and Modules, in Hazewinkel (ed.), Handbook of Algebra, vol.1.

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