Direct sum of modules

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Direct sum of modules: a short and clear guide

The direct sum is a way to combine several modules into one bigger module, with no extra relations beyond those inside each piece. It is the coproduct in the category of modules.

Two familiar settings
- Vector spaces: If V and W are vector spaces over a field K, their direct sum V ⊕ W is the set of pairs (v, w). Addition and scalar multiplication are done componentwise, and we often write elements as v + w rather than (v, w). The subspaces V × {0} and {0} × W can be identified with V and W, respectively. The dimension satisfies dim(V ⊕ W) = dim V + dim W.
- Abelian groups: If G and H are abelian groups, their direct sum G ⊕ H is G × H with componentwise operation, written g + h. The rank (free part) adds: rank(G ⊕ H) = rank(G) + rank(H).

The general case: modules over a ring
- For a family {M_i} of left R-modules indexed by I, the direct sum ⊕_{i∈I} M_i consists of all functions α from I to the union of the M_i such that α(i) ∈ M_i and α(i) = 0 for cofinitely many indices i (only finitely many terms are nonzero). If I is finite, this is the same as the direct product.
- You can also think of it as finitely supported families (α(i) nonzero for only finitely many i). Addition and scalar multiplication are done componentwise, so ⊕_{i∈I} M_i is a left R-module.

Internal direct sum and direct summands
- If M is an R-module and each M_i is a submodule of M, and every x ∈ M can be written uniquely as a finite sum of elements from the M_i, then M is the internal direct sum of the submodules M_i. In that case M is naturally isomorphic to the external direct sum ⊕_{i∈I} M_i.
- A submodule N of M is a direct summand if M = N ⊕ N′ for some submodule N′. In this situation N and N′ are complementary submodules.

The universal property (coproduct)
- The direct sum is the coproduct in the category of left R-modules. Given each i ∈ I a module M_i and an R-linear map f_i: M_i → M, there exists a unique R-linear map ⊕_{i∈I} M_i → M that makes all the diagrams commute. This universal property captures the idea that the direct sum is the most flexible way to combine the pieces.

Notes on structure and variants
- If the M_i carry extra structure (like a norm or inner product), the direct sum can be given a compatible structure, giving the coproduct in the corresponding category.
- The Grothendieck view: the direct sum makes a commutative monoid under addition; one can pass to a universal abelian group (the Grothendieck group) to allow formal "differences" of sums.

Banach and Hilbert space sums
- Banach spaces: For two Banach spaces X and Y, the direct sum X ⊕ Y uses the norm ||(x, y)|| = ||x|| + ||y||. For a family {X_i}, the ℓ^1-type direct sum consists of all functions i ↦ x_i with ∑ ||x_i|| < ∞; this is a Banach space.
- Hilbert spaces: The direct sum of Hilbert spaces H_i uses the inner product ⟨(x_i), (y_i)⟩ = ∑ ⟨x_i, y_i⟩. For finitely many summands this is straightforward; for infinitely many, take the space of all α with α(i) ∈ H_i and ∑ ||α(i)||^2 < ∞, with inner product as above, and complete if needed. For example, ℓ^2 is the direct sum of countably many copies of the real line with the usual inner product.
- Finite versus infinite: With finitely many summands, Banach and Hilbert direct sums agree up to the choice of norm; with infinitely many, they generally differ.

A final note
In the world of algebras, the term direct sum can be used differently from the category-theoretic notion of coproduct. The direct sum of algebras as vector spaces is not always the coproduct in the category of algebras. Nevertheless, the direct sum construction described here works cleanly for modules, vector spaces, abelian groups, and their many structured variants.