Radical ideals and submodules

  • Thread starter alexfloo
  • Start date
192
0
I'm currently studying commutative algebra/algebraic geometry out of Cox Little and O'Shea's Ideal Varieties and Algorithms, and linear algebra out of Steven Roman's Advanced Linear Algebra. In Roman, I'm learning about modules, and I have a question about the relationship between these two fields.

An ideal is exactly a submodule of F[x1, ..., xn] as a module over itself. We know that an ideal over this set exactly determines a subset of the affine space Fn. On the other hand, affine subsets of Fn define radical ideals of F[x,y] which are in turn submodules.

Now outwardly, the property of being radical depends on the multiplicative structure of the ideal, which the corresponding submodule doesn't have. My question is whether the property of being radical depends innately on that multiplicative structure. Do submodules corresponding to radical ideals have any identifiable properties even without recognizing their multiplicative structure?

(Wikipedia tells me that there is such a thing as a radical submodule, but I don't think it's what I'm looking for.)
 

fresh_42

Mentor
Insights Author
2018 Award
11,604
8,081
This is the reason why modules and ideals are two different things. The correspondence is only a module isomorphisms, since modules do not carry a ring structure themselves. You cannot transport the definition of a radical ideal into the module language, since multiplication is missing. But this is already true for arbitrary ideals.
 

Want to reply to this thread?

"Radical ideals and submodules" You must log in or register to reply here.

Related Threads for: Radical ideals and submodules

  • Posted
Replies
1
Views
1K
  • Posted
Replies
2
Views
2K
Replies
1
Views
542
Replies
15
Views
1K
Replies
3
Views
871
Replies
13
Views
836
Replies
5
Views
611

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top