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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 ...
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.
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 sum of two elements , ′ is defined pointwise, it is the function sending to () + ′ (), and the product is the function sending to the infinite sum , = ′ (). This sum is correctly defined (i.e., finite) because, for each m , there are only a finite number of pairs ( p , q ) such that pq = m .
The plea agreement comes weeks after Smirnov was indicted in a separate case involving federal tax charges, including concealing more than two million dollars in income he received from multiple ...
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.
"I [thought to myself], ‘Oh, no, this is not going to happen today,’ ” Linda Rosa recalled of the incident