# Decomposition Theorems

Theorem 1:
The "indecomposable" integers are the primes.

Every integer bigger than 1 is a product of powers of
indecomposables (i.e., primes).

Theorem 2:
The only "indecomposable" nontrivial real vector space is **R**.

Every nontrivial real vector space is a product of indecomposables
(i.e., **R**, **R** x **R**, etc.).

Theorem 3:
The only "indecomposable" nontrivial finite **Z**-modules are the abelian groups
**Z**_{n},

where n is a power of a prime.
Every nontrivial finite **Z**-module is a product of indecomposables.

### Some other rings have nice decomposition theorems. For more see:

R. Wiegand and S. Wiegand, *Prime ideals and decompositions of modules*, in "Non-Noetherian Commutative Ring Theory",
Kluwer, 2000, 403-428.

Studying such decompositions leads to representation theory and Dynkin diagrams.
Here are two Dynkin diagrams:

**Definition:** A *subadditive* assignment puts a positive integer in each circle so that the
number in each circle is at least twice the sum of the numbers in adjacent circles.
The assigment is *additive* if it's always exactly twice.
## Puzzle

Can you find additive assignments for each diagram (or show that none exist)?
How about assignments that are subadditive but not additive?

### For more see:

R. Wiegand and Graham Leuschke, Ascent of finite Cohen-Macaulay type, J. Algebra 228
(2000), 674-681.

## What can you do if there is no nice decomposition theorem?