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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).
If every x in M can be written in exactly one way as a sum of finitely many elements of the M i, then we say that M is the internal direct sum of the submodules M i (Halmos 1974, §18). In this case, M is naturally isomorphic to the (external) direct sum of the M i as defined above ( Adamson 1972 , p.61).
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).
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 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 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 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: