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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]).
A decomposition of a module is a way to express a module as a direct sum of submodules. dense dense submodule determinant The determinant of a finite free module over a commutative ring is the r-th exterior power of the module when r is the rank of the module. differential A differential graded module or dg-module is a graded module with a ...
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
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.
The most basic example of a semisimple module is a module over a field, i.e., a vector space. On the other hand, the ring Z of integers is not a semisimple module over itself, since the submodule 2Z is not a direct summand. Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules. Let A be an ...
So M breaks up as the direct sum of R-modules, M = e 1 M ⊕ ... ⊕ e n M. Conversely, given an R-module M 0, then M 0 ⊕n is an M n (R)-module. In fact, the category of R-modules and the category of M n (R)-modules are equivalent. The special case is that the module M is just R as a module over itself, then R n is an M n (R)-module.
Every finitely-generated R-module is a direct sum of these. Note that this is simple if and only if n = 1 (or p = 0); for example, the cyclic group of order 4, Z/4, is indecomposable but not simple – it has the subgroup 2Z/4 of order 2, but this does not have a complement. Over the integers Z, modules are abelian groups.