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:
“The burning of rebellious thoughts in the little breast, of internal hatred and opposition, could not long go on without slight whiffs of external smoke, such as mark the course of subterranean fire.”
—Harriet Beecher Stowe (1811–1896)
“Lesbian existence comprises both the breaking of a taboo and the rejection of a compulsory way of life. It is also a direct or indirect attack on the male right of access to women.”
—Adrienne Rich (b. 1929)
“And what is the potential man, after all? Is he not the sum of all that is human? Divine, in other words?”
—Henry Miller (1891–1980)