Direct Sum of Modules - Properties

Properties

• The direct sum is a submodule of the direct product of the modules M_i (Bourbaki 1989, §II.1.7). The direct product is the set of all functions α from I to the disjoint union of the modules M_i with α(i)∈M_i, but not necessarily vanishing for all but finitely many i. If the index set I is finite, then the direct sum and the direct product are equal.
• Each of the modules M_i may be identified with the submodule of the direct sum consisting of those functions which vanish on all indices different from i. With these identifications, every element x of the direct sum can be written in one and only one way as a sum of finitely many elements from the modules M_i.
• If the M_i are actually vector spaces, then the dimension of the direct sum is equal to the sum of the dimensions of the M_i. The same is true for the rank of abelian groups and the length of modules.
• Every vector space over the field K is isomorphic to a direct sum of sufficiently many copies of K, so in a sense only these direct sums have to be considered. This is not true for modules over arbitrary rings.
• The tensor product distributes over direct sums in the following sense: if N is some right R-module, then the direct sum of the tensor products of N with M_i (which are abelian groups) is naturally isomorphic to the tensor product of N with the direct sum of the M_i.
• Direct sums are also commutative and associative (up to isomorphism), meaning that it doesn't matter in which order one forms the direct sum.
• The group of R-linear homomorphisms from the direct sum to some left R-module L is naturally isomorphic to the direct product of the groups of R-linear homomorphisms from M_i to L:

  Indeed, there is clearly a homomorphism τ from the left hand side to the right hand side, where τ(θ)(i) is the R-linear homomorphism sending x∈M_i to θ(x) (using the natural inclusion of M_i into the direct sum). The inverse of the homomorphism τ is defined by

  for any α in the direct sum of the modules M_i. The key point is that the definition of τ−1 makes sense because α(i) is zero for all but finitely many i, and so the sum is finite.

  In particular, the dual vector space of a direct sum of vector spaces is isomorphic to the direct product of the duals of those spaces.

• The finite direct sum of modules is a biproduct: If

  are the canonical projection mappings and

  are the inclusion mappings, then

  equals the identity morphism of A₁ ⊕ ··· ⊕ A_n, and

  is the identity morphism of A_k in the case l=k, and is the zero map otherwise.