The "indecomposable" integers are the primes.
Every integer bigger than 1 is a product of powers of
indecomposables (i.e., primes).
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.).
The only "indecomposable" nontrivial finite Z-modules are the abelian groups
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.
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
What can you do if there is no nice decomposition theorem?