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:
“Unlike femininity, relaxed masculinity is at bottom empty, a limp nullity. While the female body is full of internal potentiality, the male is internally barren.... Manhood at the most basic level can be validated and expressed only in action.”
—George Gilder (b. 1939)
“Computer mediation seems to bathe action in a more conditional light: perhaps it happened; perhaps it didn’t. Without the layered richness of direct sensory engagement, the symbolic medium seems thin, flat, and fragile.”
—Shoshana Zuboff (b. 1951)
“No, the five hundred was the sum they named
To pay the doctor’s bill and tide me over.
It’s that or fight, and I don’t want to fight
I just want to get settled in my life....”
—Robert Frost (1874–1963)