TITLE: Thinking of rings as objects that contain ideals instead of elements
QUESTION [6 upvotes]: In this question I refer to a "commutative ring with a 1, not equal to 0" as a ring.
I have seen it mentioned a couple of times that it is more useful to think of rings as objects that contain ideals rather thinking about the elements at all. For example in some answers on thie site, they will say "here is an element-free approach". Why is this a good idea and what are the motivations behind it? From a group theory point of view it would be weird to ignore the elements, it sounds like we are just losing information. As I understand it, rings are useful because we can study number theory with them, but how could we do that element-free.
References are welcome. Thanks.

REPLY [8 votes]: Some people get carried away with the element free claims.  I'd say this is a good related question to check out.
The fact is that yes, there are sometimes useful ways of characterizing things about rings without referring to elements and only referring to ideals (or submodules of their modules). This is true for a great many things in algebraic geometry and homological algebra.
But ideals aren’t the be-all-and-end-all of ring theory.  Consider the great variety of fields (and division rings) that exist.  All of them look identical if you only pay attention to ideals.
It is true that sometimes concretely using elements results in distraction and a form of myopia.  For example, some students will only understand matrices as linear transformations at the expense of understanding linear transformations in the abstract.  In a sense, they are “too close” to see what’s going on.
But in other cases it seems impossible to avoid mentioning the elements.  For example, something like a Euclidean field needs to say something about each element.  Another example might be Boolean rings.  I’m not aware of an element free characterization of these…
Really one shouldn’t be absolutely promoting a non element approach over the element approach.  They can both be used in different circumstances, and they complement each other.