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:
“You will see Coleridgehe who sits obscure
In the exceeding lustre and the pure
Intense irradiation of a mind,
Which, with its own internal lightning blind,
Flags wearily through darkness and despair
A cloud-encircled meteor of the air,
A hooded eagle among blinking owls.”
—Percy Bysshe Shelley (17921822)
“You will find that reason, which always ought to direct mankind, seldom does; but that passions and weaknesses commonly usurp its seat, and rule in its stead.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (16941773)
“No, the five hundred was the sum they named
To pay the doctors bill and tide me over.
Its that or fight, and I dont want to fight
I just want to get settled in my life....”
—Robert Frost (18741963)