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In mathematics, a group G is called the direct sum [1] [2] of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information.
For an arbitrary family of groups indexed by , their direct sum [2] is the subgroup of the direct product that consists of the elements () that have finite support, where by definition, () is said to have finite support if is the identity element of for all but finitely many . [3] The direct sum of an infinite family () of non-trivial groups is ...
Direct sums are commutative and associative (up to isomorphism), meaning that it doesn't matter in which order one forms the direct sum. The abelian group of R-linear homomorphisms from the direct sum to some left R-module L is naturally isomorphic to the direct product of the abelian groups of R-linear homomorphisms from M i to L: (,) (,).
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 ...
The input and output domains may be the same, such as for SUM, or may be different, such as for COUNT. Aggregate functions occur commonly in numerous programming languages, in spreadsheets, and in relational algebra. The listagg function, as defined in the SQL:2016 standard [2] aggregates data from multiple rows into a single concatenated string.
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.
Trump's lawyers, citing presidential immunity and other ongoing litigation, told Merchan they oppose a hearing examining their claims of juror misconduct, and instead asked the judge to weigh the ...
A free module is a module that can be represented as a direct sum over its base ring, so free abelian groups and free -modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free -module, and each free -module comes from a free abelian group in this way. [21]