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..?