Direct Sum Of Modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces.
Read more about Direct Sum Of Modules: Construction For Vector Spaces and Abelian Groups, Construction For An Arbitrary Family of Modules, Properties, Internal Direct Sum, Universal Property, Grothendieck Group, Direct Sum of Modules With Additional Structure
Famous quotes containing the words direct and/or sum:
“Science is a system of statements based on direct experience, and controlled by experimental verification. Verification in science is not, however, of single statements but of the entire system or a sub-system of such statements.”
—Rudolf Carnap (1891–1970)
“Without doubt God is the universal moving force, but each being is moved according to the nature that God has given it.... He directs angels, man, animals, brute matter, in sum all created things, but each according to its nature, and man having been created free, he is freely led. This rule is truly the eternal law and in it we must believe.”
—Joseph De Maistre (1753–1821)