Relations
The projective indecomposable modules over some rings have very close connections with those rings' simple, projective, and indecomposable modules.
If the ring R is Artinian or even semiperfect, then R is a direct sum of principal indecomposable modules, and there is one isomorphism class of PIM per isomorphism class of simple module. To each PIM P is associated its head, P/JP, which is a simple module, being an indecomposable semi-simple module. To each simple module S is associated its projective cover P, which is a PIM, being an indecomposable, projective, cyclic module.
Similarly over a semiperfect ring, every indecomposable projective module is a PIM, and every finitely generated projective module is a direct sum of PIMs.
In the context of group algebras of finite groups over fields (which are semiperfect rings), the representation ring describes the indecomposable modules, and the modular characters of simple modules represent both a subring and a quotient ring. The representation ring over the complex field is usually better understood and since PIMs correspond to modules over the complexes using p-modular system, one can use PIMs to transfer information from the complex representation ring to the representation ring over a field of positive characteristic. Roughly speaking this is called block theory.
Over a Dedekind domain that is not a PID, the ideal class group measures the difference between projective indecomposable modules and principal indecomposable modules: the projective indecomposable modules are exactly the (modules isomorphic to) nonzero ideals and the principal indecomposable modules are precisely the (modules isomorphic to) nonzero principal ideals.
Read more about this topic: Principal Indecomposable Module
Famous quotes containing the word relations:
“I only desire sincere relations with the worthiest of my acquaintance, that they may give me an opportunity once in a year to speak the truth.”
—Henry David Thoreau (1817–1862)
“She has problems with separation; he has trouble with unity—problems that make themselves felt in our relationships with our children just as they do in our relations with each other. She pulls for connection; he pushes for separateness. She tends to feel shut out; he tends to feel overwhelmed and intruded upon. It’s one of the reasons why she turns so eagerly to children—especially when they’re very young.”
—Lillian Breslow Rubin (20th century)
“When any one of our relations was found to be a person of a very bad character, a troublesome guest, or one we desired to get rid of, upon his leaving my house I ever took care to lend him a riding-coat, or a pair of boots, or sometimes a horse of small value, and I always had the satisfaction of finding he never came back to return them.”
—Oliver Goldsmith (1728–1774)