Irreducible Components in a Primary Decomposition

  • Thread starter ircdan
  • Start date
  • #1
229
0

Main Question or Discussion Point

Let R be a Noetherian Ring and I an ideal in R.

Let I = Q_1 n ... n Q_r = J_1 n ... n J_r be two primary decompositions of I.

How can I show the number of irreducible components in each decomposition is the same?
 

Answers and Replies

  • #2
matt grime
Science Advisor
Homework Helper
9,395
3
Firstly you shoulnd't have r on both sides of that equality. Secondly, you will need to know the definition of 'primary'. So what is it? What does it imply? Does this seem familiar to any other result you know? Is the result even true? (i.e. think about what a counter example might be and see where it fails to satisfy some hypothesis.)
 
  • #3
229
0
Firstly you shoulnd't have r on both sides of that equality. Secondly, you will need to know the definition of 'primary'. So what is it? What does it imply? Does this seem familiar to any other result you know? Is the result even true? (i.e. think about what a counter example might be and see where it fails to satisfy some hypothesis.)
Matt, thank you very much for your reply. Yes the result is true. I'm assuming all uniqueness theorems found in say, Dummit and Foote. I figured it would be easier to prove with some theory in place, hence why I have r on both sides. Whether I need this or not I don't know, but I figured it might make things easier.

Any help would be greatly appreciated.
 
Last edited:
  • #4
matt grime
Science Advisor
Homework Helper
9,395
3
The point was that by writing r on both sides you were assuming the result you wanted to show. And you haven't written out the definition of primary/prime and what it implies in a Noetherian ring. The result will be deducible from the definitions, so what are the definitions? I gave you plenty of help and you ignored it all.
 
  • #5
mathwonk
Science Advisor
Homework Helper
10,820
985
i think you need also to assume the decompositions are irredundant, or it is not true.
 

Related Threads on Irreducible Components in a Primary Decomposition

  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
3
Views
6K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
4
Views
2K
Replies
6
Views
818
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
2K
Top