Indecomposable module
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In abstract algebra, a module is defined to be indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. In many situations, all modules of interest can be written as direct sums of indecomposable ones; the indecomposable modules can then be thought of as the "basic building blocks", the only objects that need to be studied.
Examples
The modules over fields are vector spaces. A vector space is indecomposable if and only if its dimension is 1. So every vector space is a direct sum of indecomposable ones (with infinitely many summands if the dimension is infinite).
The modules over the ring of integers Z are the abelian groups. An abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form Z/pnZ for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of (finitely many) indecomposable abelian groups. There are, however, abelian groups that cannot be written as a (finite or infinite) direct sum of indecomposables; the rational numbers Q form the simplest example.
The above situation over Z can be generalized in a straight-forward manner to modules over any principal ideal domain R: the indecomposable modules are R and R/pnR for prime ideals p in R. Every finitely-generated R-module is a direct sum of these.
For a fixed positive integer n, consider the ring R of n-by-n matrices with entries from the real numbers (or from any other field K). Then Kn is a left R-module (the scalar multiplication is matrix multiplication). This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of (finitely or infinitely many) copies of this module Kn.
Facts
Every simple module is indecomposable. The converse is not true in general, as is shown by the second example above.
By looking at the endomorphism ring of a module, one can tell whether the module is indecomposable: if and only if the endomorphism ring contains an idempotent different from 0 and 1. (If f is such an idempotent endomorphism of M, then M is the direct sum of ker(f) and im(f).)
A module of finite length is indecomposable if and only if its endomorphism ring is local. Still more information about endomorphisms of finite-length indecomposables is provided by the Fitting lemma.
In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull-Schmidt theorem: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique (meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic).