Understanding in abstract algebra

[mordechai9]

In algebra, do you just base your understanding off the pure definitions and groups? I am learning some multilinear algebra, seeing a lot of talk about rings, algebras, modules, etc. and I can't help but thinking it's all just frivolous, pointless definitions. That's partly because I just can never seem to "lock on" to a real understanding of things... I always forget the definitions of algebra and ring even though I see them over and over, whereas I remember other definitions (e.g. Leibniz rule, chain rule, etc.) a bit easier... I know there is a common split between algebra/analysis people but this doesn't shed light on the problem for me. Why am I incapable of understanding and appreciating all this algebra?

In short, to end my rambling, what is going on with this? Comments?

 

[eok20]

Like all things in math, the best way to get to grips with these objects are to work out problems with specific rings, algebras, modules, etc. I remember when I was first learning this stuff I kept trying to memorize what the axioms for a group, ring, etc were and it felt very unnatural. However, after working with these objects a lot you'll pick up a feel for their character, which sheds light on the definitions and makes you remember what they are. For example, if you work through enough proofs in group theory, you will know how important it is that inverses exist.

Your examples of rings, algebras, and modules can be a little confusing because there's so much overlap-- for example, an algebra is also a ring and a module. But each of these has their own character, for example that an algebra is a vector space (instead of any old module) makes it a lot easier to work with (for example, you can talk about dimension and work with bases) than general modules. In my experience, it is easiest to think of modules as general vector spaces but remind yourself that not everything that works for vector spaces works for modules.

 

[The Chaz]

Just a comment...
The chain rule isn't a definition; it's a result of definitions. Please understand: I'm not trying to be belligerent. This is an important distinction to make and to understand. We can "define" the dot product of two vectors (if you aren't familiar with this, it's still worth learning as it is easy, visible, and intuitive), but it is a RESULT of this definition that perpendicular vectors have a dot product of zero.

 

[mordechai9]

Chaz -- you're right, sorry, chain rule is obviously a result not a definition, I was just speaking crudely.

 

[The Chaz]

Chaz -- you're right, sorry, chain rule is obviously a result not a definition, I was just speaking crudely.

It doesn't bother me in the slightest. I can totally sympathize with your frustrations, and have been at the "I don't give a #&^% about PRECISION!" stage about... 1000 times in the past 4 months alone! Not that you would resort to such passion...

I believe that definitions are created, in general, out of necessity. Smart guys do a lot of imagining and realize that they need to formalize.

 

[Martin Rattigan]

Probably best not to get involved in too many at once. Start with groups and do lots of exercises. Then go onto rings etc.

 

[Monocles]

You'll develop intuition for it over time. I'm more of a geometrically-inclined person, and I too found groups to be a collection of meaningless definitions at first (luckily this was before I took abstract algebra!). However, I gained intuition for how groups work by learning about Lie groups, which are groups that are also differentiable manifolds. You don't even need to know what a differentiable manifold is! Probably the most accessible non-trivial Lie group to look at is SO(2), the group of rotations in 2 dimensions.

Once you get an intuitive understanding of what Lie groups are like, you might be more prepared to attack discrete groups. This is what happened to me. I found it especially helpful when learning about normal subgroups - you can *visualize* a normal subgroup of a Lie group.

This approach might not be helpful to everyone, but I certainly found it useful. I have not found a similar way to think about other algebraic structures - but once you're used to groups moving onto other structures is easier.