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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]).
An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there is a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both.
A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5] Semisimple A semisimple module is a direct sum (finite or not) of simple modules.
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.
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.
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The direct sum of two nonzero uniform modules always contains two submodules with intersection zero, namely the two original summand modules. If N 1 and N 2 are proper submodules of a uniform module M and neither submodule contains the other, then M / ( N 1 ∩ N 2 ) {\displaystyle M/(N_{1}\cap N_{2})} fails to be uniform, as