# Irreducible Components in a Primary Decomposition

• ircdan
In summary, the conversation discusses two primary decompositions of an ideal in a Noetherian ring and how to show that the number of irreducible components in each decomposition is the same. The speaker suggests knowing the definition of 'primary' and its implications, and considering possible counterexamples to prove the result. Additionally, assuming the decompositions are irredundant may be necessary for the result to hold.
ircdan
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?

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 grime said:
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:
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.

i think you need also to assume the decompositions are irredundant, or it is not true.

## 1. What are irreducible components in a primary decomposition?

Irreducible components are the building blocks of a primary decomposition. They are the minimal nonempty closed subsets of a topological space that cannot be written as a union of two proper closed subsets.

## 2. How are irreducible components related to primary ideals?

Irreducible components correspond to primary ideals in a primary decomposition. Each primary ideal corresponds to a unique irreducible component, and vice versa.

## 3. Why are irreducible components important in primary decomposition?

Irreducible components play a crucial role in primary decomposition because they provide a way to decompose a complex topological space into simpler, more manageable parts. This allows for a better understanding and analysis of the space.

## 4. Can a topological space have multiple primary decompositions?

Yes, a topological space can have multiple primary decompositions. However, the number of irreducible components will remain the same in each decomposition.

## 5. How are irreducible components determined in a primary decomposition?

The irreducible components in a primary decomposition can be determined using the Nullstellensatz theorem in algebraic geometry or the Prime Ideal Decomposition theorem in commutative algebra.

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