# Irreducible Components in a Primary Decomposition

## 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?

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matt grime
Homework Helper
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.)

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:
matt grime