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:
“Get your facts first, and then you can distort them as much as you please.”
—Mark Twain [Samuel Langhorne Clemens] (18351910)
“In writing biography, fact and fiction shouldnt be mixed. And if they are, the fictional points should be printed in red ink, the facts printed in black ink.”
—Catherine Drinker Bowen (18971973)
“A pun does not commonly justify a blow in return. But if a blow were given for such cause, and death ensued, the jury would be judges both of the facts and of the pun, and might, if the latter were of an aggravated character, return a verdict of justifiable homicide.”
—Oliver Wendell Holmes, Sr. (18091894)