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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.
If is a module that satisfies the ACC and DCC on submodules (that is, it is both Noetherian and Artinian or – equivalently – of finite length), then is a direct sum of indecomposable modules. Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism.
Then the left R-module R is isomorphic to the left R-module R ⊕ R. Example: (Lam 2003, Exercise 8.16) If A and B are any groups then the Eilenberg swindle can be used to construct a ring R such that the group rings R[A] and R[B] are isomorphic rings: take R to be the group ring of the restricted direct product of infinitely many copies of A ...
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)} .
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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 ...