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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 direct sum is also commutative up to isomorphism, i.e. for any algebraic structures and of the same kind. The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups.
An indecomposable module is a module that is not a direct sum of two nonzero submodules. Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism rings , then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in group theory , is known as ...
A graded module is a module with a decomposition as a direct sum M = ⨁ x M x over a graded ring R = ⨁ x R x such that R x M y ⊆ M x+y for all x and y. Uniform A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.
A direct sum of modules is a module that is the direct sum of the underlying abelian group together with component-wise scalar multiplication. dual module The dual module of a module M over a commutative ring R is the module Hom R ( M , R ) {\displaystyle \operatorname {Hom} _{R}(M,R)} .
In mathematics, a group G is called the direct sum [1] [2] of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information.
Torsionfree modules over a Dedekind domain are determined (up to isomorphism) by rank and Steinitz class (which takes value in the ideal class group), and the decomposition into a direct sum of copies of R (rank one free modules) is replaced by a direct sum into rank one projective modules: the individual summands are not uniquely determined ...
The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded -algebra.