Internal Direct Sum
See also: Internal direct productSuppose M is some R-module, and Mi is a submodule of M for every i in I. If every x in M can be written in one and only one way as a sum of finitely many elements of the Mi, then we say that M is the internal direct sum of the submodules Mi (Halmos 1974, §18). In this case, M is naturally isomorphic to the (external) direct sum of the Mi as defined above (Adamson 1972, p.61).
A submodule N of M is a direct summand of M if there exists some other submodule N′ of M such that M is the internal direct sum of N and N′. In this case, N and N′ are complementary subspaces.
Read more about this topic: Direct Sum Of Modules
Famous quotes containing the words internal, direct and/or sum:
“When a person doesn’t understand something, he feels internal discord: however he doesn’t search for that discord in himself, as he should, but searches outside of himself. Thence a war develops with that which he doesn’t understand.”
—Anton Pavlovich Chekhov (1860–1904)
“Authority and power are two different things: power is the force by means of which you can oblige others to obey you. Authority is the right to direct and command, to be listened to or obeyed by others. Authority requests power. Power without authority is tyranny.”
—Jacques Maritain (1882–1973)
“[M]y conception of liberty does not permit an individual citizen or a group of citizens to commit acts of depredation against nature in such a way as to harm their neighbors and especially to harm the future generations of Americans. If many years ago we had had the necessary knowledge, and especially the necessary willingness on the part of the Federal Government, we would have saved a sum, a sum of money which has cost the taxpayers of America two billion dollars.”
—Franklin D. Roosevelt (1882–1945)