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 does not contain 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).
Read more about this topic: Indecomposable Module
Famous quotes containing the word facts:
“Perhaps the facts most astounding and most real are never communicated by man to man.”
—Henry David Thoreau (1817–1862)
“The bases for historical knowledge are not empirical facts but written texts, even if these texts masquerade in the guise of wars or revolutions.”
—Paul Deman (1919–1983)
““Now what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of service to them.””
—Charles Dickens (1812–1870)