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The xy-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces, namely the x and y axes. In this direct sum, the x and y axes intersect only at the origin (the zero vector).
The Jordan identity implies that if x and y are elements of A, then the endomorphism sending z to x(yz)−y(xz) is a derivation. Thus the direct sum of A and der(A) can be made into a Lie algebra, called the structure algebra of A, str(A). A simple example is provided by the Hermitian Jordan algebras H(A,σ).
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 given direct sum decomposition of into complementary subspaces still specifies a projection, and vice versa. If X {\displaystyle X} is the direct sum X = U ⊕ V {\displaystyle X=U\oplus V} , then the operator defined by P ( u + v ) = u {\displaystyle P(u+v)=u} is still a projection with range U {\displaystyle U} and kernel V {\displaystyle V} .
The direct sum of infinitely many free abelian groups remains free abelian. It has a basis consisting of tuples in which all but one element is the identity, with the remaining element part of a basis for its group. [8] Every free abelian group may be described as a direct sum of copies of , with one copy for each member of its basis. [13] [14 ...
A graded module is a module with a decomposition as a direct sum M = ⨁ x M x over a graded ring R = ⨁ x R x such that R x M y ⊆ M x+y for all x and y. Uniform A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.
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In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the ...