Direct sum of modules

The direct sum is a construction which combines several vector spaces (or abelian groups, or modules) into a new, bigger one. In a sense, the direct sum of vectorspace is the "most general" vector space that contains the given ones as subspaces.

Construction for two vector spaces

Suppose V and W are vector spaces over the field K. We can turn the cartesian product V × W into a vector space over K by defining the operations componentwise:

• (v₁, w₁) + (v₂, w₂) = (v₁ + v₂, w₁ + w₂)
• α (v, w) = (α v, α w)

The resulting vector space is called the direct sum of V and W and is usually denoted by V oplus W, where 'oplus' is a plus symbol inside a circle.

The subspace V × {0} of V oplus W is isomorphic to V and is often identified with V; similar for {0} × W and W. With this identification, it is true that every element of V oplus W can be written in one and only one way as the sum of an element of V and an element of W. The dimension of V oplus W is equal to the sum of the dimensions of V and W.

Construction for arbitrary many modules

The direct sum can also be defined for abelian groups and for modules over arbitrary rings. Note that abelian groups are modules over the ring Z of integers, and vector spaces are modules over fields. So we only need to consider the case of modules in the sequel.

Assume R is some ring, I some set, and for every i in I we are given a left R-module M_i. The direct sum of these modules is then defined to be the set of all functions α with domain I such that α(i) ∈ M_i for all i ∈ I and α(i) = 0 for all but finitely many indices i.

Two such functions α and β can be added by writing (α + β)(i) = α(i) + β(i) for all i (note that this is again zero for all but finitely many indices), and such a function can be multiplied with an element r from R by writing (rα)(i) = r(α(i)) for all i. In this way, the direct sum becomes a left R module. We denote it by Oplus_i∈I M_i.

Properties

With the proper identifications, we can again say that every element x of the direct sum can be written in one and only one way as a sum of finitely many elements of the 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.

In the language of category theory, the direct product is a coproduct in the category of left R-modules, which means that it is characterized by the following universal property. For every i in I, consider the natural embedding j_i : M_i -> Oplus_i∈I M_i which sends the elements of M_i to those functions which are zero for all arguments but i. If f_i : M_i -> M are arbitrary R-linear maps for every i, then there exists precisely one R-linear map f : Oplus_i∈I M_i -> M such that f o j_i = f_i for all i.

Internal direct sums

Suppose M is some R-module, and M_i is a submodule of M for every i in I. If every x in M can be written in one and only 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. In this case, M is naturally isomorphic to the (external) direct sum of the M_i as defined above.