Approaches for the study of rings

Question

When, as a student in my first weeks, I first read the definition of a group, the concept of groups looked very natural and motivated to me. But when I read the definition of a ring some time later, I had no idea what rings should be about and how they were motivated. Especially the distributive property let me wonder how much of the multiplicative structure is already determined by the addition.

All of this is has been two years ago. When I recently started to read Algebra: Chapter $\it 0$ by Paolo Aluffi, a lot of thinks I had not understood before, became clear to me. Mostly because of the use of categorical language which really works well with the way I think about mathematics.
I even came to peace with the concept of rings, when Aluffi explained them as coming up in the context of studying the homsets of abelian groups (we take the group operation as addition and the composition as multiplication).
I have been very happy about this explanation, but then, instead of studying ring by using this approach, Aluffi used the 'classical' definition of rings again.

My question is: Why?

Is it not much more promising to study rings by studying the homsets of abelian groups? I know that there is not any loose of information by just using the classical definition, but using the definition via homsets looks a lot more natural to me.
I know some category theory (at least I read Emily Rhiels book Category Theory in Context) so I could read I bit trough nLab and soon found out that a ring is equivalent to a preadditive category with only one object (encapturing pretty much what I called 'homset definiton'). So why don't we study thous preadditive categories instead? They seem to have far more 'visible structure' then rings have in their usual definition.

(I know that this is kind of a soft question. I do not look in particular for a kind of 'rigorous answer', but rather for your personal view towards this.)

Answer 1

I wanted to expand on my comment, but my expansion was getting too big, so I've decided to add it as an answer. Just like the question, my answer is somewhat soft (and is certainly opinionated) so take it with a grain of salt.

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Every group is a subgroup of a symmetric group - this doesn't mean that we should only study symmetric groups. Similarly, every ring is a subring of the endomorphism ring of an abelian group, but there is good reason to not restrict ourselves to this case.

Similarly, wrt the "categorical" definitions. Aluffi presents (somewhat cheekily)

Joke 1.1: A group is a groupoid with a single object.

On the very next page, Aluffi gives "Definition 1.2" which is the traditional definition of a group. I'm sure he could have included a similar joke regarding the definition of rings (instead of giving homsets as an example of rings), but doubtless he would have also settled on the "traditional" definition of rings as his real definition.

Now, to try to address your question of "why":

The categorical definitions are nice in that they let us see relations between structures, and often give us tools for proving something in multiple areas simultaneously by abstracting almost everything specific away, leaving only the structure of the problem.

This can be useful, but only after having seen the "traditional" definitions of our objects. First and foremost, the traditional definitions require no background knowledge. When we give the definition of a ring, our examples can be things like $\mathbb{Z}$ and $\mathbb{Q}[x]$. Extremely concrete objects that we have been playing with since middle school. When we give the definition of a ring in terms of homsets of abelian groups, suddenly we don't have our concrete examples anymore, or at least, we don't obviously have them. This is pedagogically worse, as it obscures why rings are the way they are -- because they generalize things we already care about.

Not to mention the categorical definitions often presuppose you know some category theory! To say that "a group is a groupoid with one element" is nice and all, but it's only helpful if your audience understands groupoids! When introducing a new topic, we want to draw analogies between the new topic and things the audience already knows. Since categorical tools are at the top of the abstraction hierarchy, the intuition we get for, say, homsets, comes from our knowledge of rings. Not the other way around. This is because rings are "closer" to simple things like $\mathbb{Z}$ than homsets are.

I agree that rings are, at face value, the grossest of the "big three" algebraic structures (groups/rings/fields). But through the study of their modules, and eventually through algebraic geometry, I learned to love them (though noncommutative rings still scare me...).

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The moral is that the traditional definition is traditional for a reason, and trying to look for abstraction too soon is likely to confuse rather than enlighten. To learn to love rings, you just need to spend some more time with them, on their own terms. They arise very naturally in algebraic geometry -- perhaps that is a good place to look for your justification. At the very least, I hope you can see that, even if the categorical definition is better for you (which I'm still not convinced it is), the given definition is likely to be more useful for more people.

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I hope this helps ^_^

Answer 2

Just to add:

We have the four basic operations of numbers: $+,\, -,\,\cdot,\,/$.

It's thus natural to define fields as abstract algebraic structures in which certain essential properties of these operations hold.

Then we can loosen it: if we drop commutativity of multiplication, we get skew fields (aka. division rings).

If we drop division (i.e. multiplicative inverses are not assumed), we get rings.
Rings (with $+,\, -, \, \cdot$) are able to do number theory in an abstract level by talking about divisibility, factorization. E.g. according to Dedekind, ideals represent 'ideal divisors' which in general might not be represented by an element of the ring.

If we drop multiplication, we get Abelian groups.

If we drop commutativity (or drop $+,-, 0$ from the definition of a skew field), we receive groups.