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:
“I have been amazed by the Anglo-Saxon’s lack of curiosity about the internal lives and emotions of the Negroes, and for that matter, any non-Anglo-Saxon peoples within our borders, above the class of unskilled labor.”
—Zora Neale Hurston (1891–1960)
“No direct hit to smash the shatter-proof
And lodge at last the quivering needle
Clean in the eye of one who stands transfixed
In fascination of her brightness.”
—Karl Shapiro (b. 1913)
“To sum up:
1. The cosmos is a gigantic fly-wheel making 10,000 revolutions a minute.
2. Man is a sick fly taking a dizzy ride on it.
3. Religion is the theory that the wheel was designed and set spinning to give him the ride.”
—H.L. (Henry Lewis)