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The direct sum of abelian groups is a prototypical example of a direct sum. Given two such groups ( A , ∘ ) {\displaystyle (A,\circ )} and ( B , ∙ ) , {\displaystyle (B,\bullet ),} their direct sum A ⊕ B {\displaystyle A\oplus B} is the same as their direct product .
The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
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.
Unlike a finite direct product, the infinite direct product Π i∈I G i is not generated by the elements of the isomorphic subgroups { G i } i∈I. Instead, these subgroups generate a subgroup of the direct product known as the infinite direct sum, which consists of all elements that have only finitely many non-identity components.
For example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms. (It therefore coincides ...
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
A decomposition with local endomorphism rings [5] (cf. #Azumaya's theorem): a direct sum of modules whose endomorphism rings are local rings (a ring is local if for each element x, either x or 1 − x is a unit). Serial decomposition: a direct sum of uniserial modules (a module is uniserial if the lattice of submodules is a finite chain [6]).
One example is R/I where R is a direct product of countably many copies of F 2 and I is the direct sum of countably many copies of F 2 inside of R. The R-module R/I is locally free since R is Boolean (and it is finitely generated as an R-module too, with a spanning set of size 1), but R/I is not projective because I is not a principal ideal.