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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.
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 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 ...
The former may be written as a direct sum of finitely many groups of the form / for prime, and the latter is a direct sum of finitely many copies of . If f , g : G → H {\displaystyle f,g:G\to H} are two group homomorphisms between abelian groups, then their sum f + g {\displaystyle f+g} , defined by ( f + g ) ( x ) = f ( x ) + g ( x ...
The Kronecker sum is different from the direct sum, but is also denoted by ⊕. It is defined using the Kronecker product ⊗ and normal matrix addition. If A is n -by- n , B is m -by- m and I k {\displaystyle \mathbf {I} _{k}} denotes the k -by- k identity matrix then the Kronecker sum is defined by:
Holiday get-togethers often feature a lot of alcohol. If you’re trying to reduce your intake, here’s how to plan ahead and still have a celebratory holiday season.
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Given an ideal I in a commutative ring R and an R-module M, the direct sum = / + is a graded module over the associated graded ring / +. A morphism f : N → M {\displaystyle f:N\to M} of graded modules, called a graded morphism or graded homomorphism , is a homomorphism of the underlying modules that respects grading; i.e., f ( N i ) ⊆ M ...