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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.

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rdf:type	Thing company
rdfs:label	Direct sum of modules (en)
rdfs:comment	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. (en)
rdfs:seeAlso	Direct product
sameAs	Direct sum of modules Direct sum of modules
dbp:wikiPageUsesTemplate	dbt:Anchor dbt:Annotated_link dbt:Citation dbt:Em dbt:For dbt:Further dbt:Harv dbt:Reflist dbt:See_also dbt:Pb
Subject	Linear algebra Module theory
gold:hypernym	Construction
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Direct_sum_of_modules?oldid=1074041511&ns=0
Wikipage page ID	58899 (xsd:integer)
page length (characters) of wiki page	22590 (xsd:nonNegativeInteger)
Wikipage revision ID	1074041511 (xsd:integer)
Link from a Wikipage to another Wikipage	Grothendieck group Norm (mathematics) Product (category theory) Support (mathematics) William Kingdon Clifford Abelian group Abstract algebra Algebra over a field Algebraic structure Associative property Banach space Bicomplex number Bilinear form Biproduct Monoid Sequence space Universal property Cartesian product Category (mathematics) Category theory James Cockle Joseph Wedderburn Linear algebra Module theory Dimension (vector space) Direct product Direct sum Direct sum of modules Disjoint union Domain of a function Dual (category theory) Dual space Hilbert space Length of a module Set (mathematics) Decomposition of a module Hypercomplex number Linear map Split-biquaternion Split-complex number Tensor product Tensor product of algebras Cofiniteness Commutative property Complemented subspace Coproduct Homomorphism Orthogonal complement Orthogonality Finite set Function (mathematics) Ian R. Porteous Index set Indexed family Fiber bundle Field (mathematics) Inner product space Integer Interval arithmetic Module (mathematics) Vector space Subgroup Limit (category theory) Pointwise Rank of an abelian group Ring homomorphism dbr:Lindenstrauss–Tzafriri_theorem
has abstract	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. See the article decomposition of a module for a way to write a module as a direct sum of submodules. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Direct_sum_of_modules
is Wikipage redirect of	Direct sum of Lie algebras Direct sum of algebras Direct sum of Banach spaces Direct sum of Hilbert spaces Direct sum of vector spaces Orthogonal direct sum Complementary subspace

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