Commutative Algebra: Study of Rings & Groups

In summary: It seems like there is a lot of interest in this topic.In summary, the motivation for commutative algebra is the study of commutative groups, which are generally not very exciting. However, if one considers modules over the ring Z, then one gets more results.
  • #1
pivoxa15
2,255
1
Why is CA mostly to do with the study of rings? Why not study more commutative groups? Or are most group noncommutative?
 
Physics news on Phys.org
  • #2
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.
 
  • #3
Why not study commutative groups by themselves? Do they not occur often enough?
 
  • #4
Who's stopping you?
 
  • #5
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.
 
  • #6
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).
 
Last edited:
  • #7
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.
 
  • #8
CompuChip said:
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.
 

1. What is the definition of a ring in commutative algebra?

A ring in commutative algebra is a set equipped with two operations, addition and multiplication, that satisfy certain properties. These properties include closure under addition and multiplication, associativity, commutativity of addition, the existence of an additive identity element, and the distributive property.

2. How does commutative algebra differ from other branches of algebra?

Commutative algebra focuses specifically on commutative rings and groups, which have the property that the order of multiplication does not affect the result. This differs from other branches of algebra, such as non-commutative algebra, which allow for the possibility of non-commutativity in their structures.

3. What are some real-world applications of commutative algebra?

Commutative algebra has applications in a wide range of fields, including cryptography, coding theory, algebraic geometry, and number theory. It is also used in computer science and physics, particularly in the study of symmetry and group theory.

4. Can you explain the concept of a polynomial ring in commutative algebra?

A polynomial ring in commutative algebra is a ring formed by adding a set of variables to a given ring and defining multiplication in a specific way. These variables can be thought of as unknowns, and the multiplication is defined such that it follows the usual rules of polynomial multiplication, such as the distributive property and the rule for multiplying powers of the same variable.

5. How does the study of rings and groups in commutative algebra relate to other areas of mathematics?

Commutative algebra has connections to many other areas of mathematics, such as algebraic geometry, number theory, and topology. It also has applications in areas such as algebraic coding theory, homological algebra, and representation theory. The study of rings and groups in commutative algebra provides a foundation for understanding these other areas and can be used to solve problems in these fields.

Similar threads

Replies
24
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
16
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
Replies
2
Views
706
  • Linear and Abstract Algebra
Replies
3
Views
690
  • Linear and Abstract Algebra
Replies
1
Views
600
  • Linear and Abstract Algebra
Replies
6
Views
1K
  • Linear and Abstract Algebra
Replies
9
Views
1K
Back
Top