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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.
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]).
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.
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)} .
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 coproduct in the category of sets is simply the disjoint union with the maps i j being the inclusion maps.Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be ...
The biproduct is again the direct sum, and the zero object is the trivial vector space. More generally, biproducts exist in the category of modules over a ring. On the other hand, biproducts do not exist in the category of groups. [4] Here, the product is the direct product, but the coproduct is the free product.
For a finite projective module over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma. [3] For the general case, the proof (both the original as well as later one) consists of the following two steps: Observe that a projective module over an arbitrary ring is a direct sum of countably generated projective modules.