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:
“Neither moral relations nor the moral law can swing in vacuo. Their only habitat can be a mind which feels them; and no world composed of merely physical facts can possibly be a world to which ethical propositions apply.”
—William James (1842–1910)
“Each truth that a writer acquires is a lantern, which he turns full on what facts and thoughts lay already in his mind, and behold, all the mats and rubbish which had littered his garret become precious. Every trivial fact in his private biography becomes an illustration of this new principle, revisits the day, and delights all men by its piquancy and new charm.”
—Ralph Waldo Emerson (1803–1882)
““It is of the highest importance in the art of detection to be able to recognise out of a number of facts which are incidental and which are vital.... I would call your attention to the curious incident of the dog in the night-time.”
“The dog did nothing in the night-time.”
“That was the curious incident.””
—Sir Arthur Conan Doyle (1859–1930)