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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.

user969530
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    Where have you seen this statement? It does make some kind of sense in applications of ring theory such as algebraic geometry, but we can't give you any specific help it you don't give specific references. – Rob Arthan Sep 18 '21 at 20:17

2 Answers2

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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.

rschwieb
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This is a bit opinion based, but I'll try to answer it anyway.

The motivation is that sometimes too much information is just a noise that blurs the big picture. For example if $R$ has exactly two ideals ($0$ and $R$) then it is a field. And so we can say alot about modules over $R$ (all are free) simply by looking at how many ideals $R$ has, no need to analyze elements. Of course the same could be deduced by checking whether every element has inverse, after all having more information doesn't make things worse. But we are just humans and there's limited amount of information we can handle.

Note that the same applies to groups. We often consider simple groups, which are groups with only two normal subgroups: the trivial subgroup and itself. This information alone implies a lot of consequences, like for example every homomorphism with a simple group as a domain is either zero or injective. In fact finite simple groups are very important, they are building blocks of any finite group.

Another example, say you have a concrete group $G$. The group is big, the multiplication table is complicated, but somehow you've managed to deduce that it has precisely two subgroups: the trivial subgroup and itself. Voilà, this is enough to deduce that $G$ is a cyclic group of prime order. No need to find the generator and calculate its order. Unless it is needed of course.

freakish
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