Commutative Algebra: Study of Rings & Groups

[pivoxa15]

Why is CA mostly to do with the study of rings? Why not study more commutative groups? Or are most group noncommutative?

 

[mathwonk]

yes. commutative groups are in fact the motivation for commutative algebra. i.e. commutative groups are interpreted as modules over the commutative ring Z. in general, we consider modules over commutative rings.

 

[pivoxa15]

Why not study commutative groups by themselves? Do they not occur often enough?

 

[morphism]

Who's stopping you?

 

[pivoxa15]

CA is an established subject and I don't know any of it yet. Just like to know why the experts don't consider groups (without rings attached) in CA much.

 

[CompuChip]

The Wikipedia page says that "infinite abelian groups are the subject of current research" (you never know how true information on Wikipedia is though, especially on such statements).

Though I usually don't have much to do with group theory, one reason I can think of for not studying commutative groups a lot, is that any subgroup is normal. Now I understand that usually we try to describe larger, new groups by studying their normal subgroups and the corresponding quotients. I can imagine, that commutative groups are not really exciting in this respect: just keep dividing out subgroups until you have reduced it to small pieces, all of which you know.

Also, I think you are right in the remark in your first post: AFAIK commutativity is quite a special property for a group to have, and most groups will be non-commutative anyway (for example, the rotation group in two dimensions is commutative, but not very exciting -- if you want to study rotations in three dimensions you already lose commutativity).

 

[mathwonk]

there is no objection to studying commutative groups, but the point of commutative algebra is to widen the scope of what is known about commutative groups so thT IT Applies to more situations. i.e. given a commutative group that is finitely generated, one has a lot of strong results.

but what about a commutative group that is not finitely generated? if we look at it over a different ring, then maybe it will be finitely generated over that? that let's us apply results that were previously only available for finitely generated groups.

so by lifting your gaze above the restrictions of abelian groups, to consider modules, one gets more results.

 

[quasar987]

The Wikipedia page says that "infinite abelian groups are the subject of current research" (you never know how true information on Wikipedia is though, especially on such statements).

My current algebra professor indeed works on the theory of infinite abelian groups. And so does another professor at my university.