Search results
Results from the WOW.Com Content Network
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.
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 ...
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]).
In more technical language, if the summands are (), the direct sum is defined to be the set of tuples () with such that = for all but finitely many i. The direct sum is contained in the direct product, but is strictly smaller when the index set is infinite, because an element of the direct product can have infinitely many nonzero coordinates.
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.
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 ...
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
An R-module M is semi-simple if every R-submodule of M is an R-module direct summand of M (the trivial module 0 is semi-simple, but not simple). For an R -module M , M is semi-simple if and only if it is the direct sum of simple modules (the trivial module is the empty direct sum).