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..?
 A: 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.)
A: 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.


