Direct sum

A distinction is made between internal and external direct sums, though the two are isomorphic. If the factors are defined first, and then the direct sum is defined in terms of the factors, we have an external direct sum. For example, if we define the real numbers ${\displaystyle \mathbb {R} }$ and then define ${\displaystyle \mathbb {R} \oplus \mathbb {R} }$ the direct sum is said to be external.

If, on the other hand, we first define some algebraic structure ${\displaystyle S}$ and then write ${\displaystyle S}$ as a direct sum of two substructures ${\displaystyle V}$ and ${\displaystyle W}$, then the direct sum is said to be internal. In this case, each element of ${\displaystyle S}$ is expressible uniquely as an algebraic combination of an element of ${\displaystyle V}$ and an element of ${\displaystyle W}$. For an example of an internal direct sum, consider ${\displaystyle \mathbb {Z} _{6}}$ (the integers modulo six), whose elements are ${\displaystyle \{0,1,2,3,4,5\}}$. This is expressible as an internal direct sum ${\displaystyle \mathbb {Z} _{6}=\{0,2,4\}\oplus \{0,3\}}$.

The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups ${\displaystyle (A,\circ )}$ and ${\displaystyle (B,\bullet )}$, their direct sum ${\displaystyle A\oplus B}$ is the same as their direct product, that is the underlying set is the Cartesian product ${\displaystyle A\times B}$ and the group operation ${\displaystyle \cdot }$ is defined component-wise:

${\displaystyle (a_{1},b_{1})\cdot (a_{2},b_{2})=(a_{1}\circ a_{2},b_{1}\bullet b_{2})}$.

This definition generalizes to direct sums of finitely many abelian groups.

For an infinite family of abelian groups A_i for i ∈ I, the direct sum

${\displaystyle \bigoplus _{i\in I}A_{i}}$

is a proper subgroup of the direct product. It consists of the elements ${\displaystyle \textstyle (a_{i})\in \prod _{j\in I}A_{j}}$ such that a_i is the identity element of A_i for all but finitely many i.[2]

The direct sum of modules is a construction which combines several modules into a new module.

The most familiar examples of this construction occur when considering vector spaces, which are modules over a field. The construction may also be extended to Banach spaces and Hilbert spaces.

The direct sum of group representations generalizes the direct sum of the underlying modules, adding a group action to it. Specifically, given a group G and two representations V and W of G (or, more generally, two G-modules), the direct sum of the representations is V ⊕ W with the action of g ∈ G given component-wise, i.e.

g·(v, w) = (g·v, g·w).

Another equivalent way of defining the direct sum is as follows:

Given two representations ${\displaystyle (V,\rho _{V})}$ and ${\displaystyle (W,\rho _{W})}$ the vector space of the direct sum is ${\displaystyle V\oplus W}$ and the homomorphism ${\displaystyle \rho _{V\oplus W}}$ is given by ${\displaystyle \alpha \circ (\rho _{V}\times \rho _{W})}$, where ${\displaystyle \alpha :GL(V)\times GL(W)\to GL(V\otimes W)}$ is the natural map obtained by coordinate-wise action as above.

Furthermore, if ${\displaystyle V,\,W}$ are finite dimensional, then, given a basis of ${\displaystyle V,\,W}$, ${\displaystyle \rho _{V}}$ and ${\displaystyle \rho _{W}}$ are matrix-valued. In this case, ${\displaystyle \rho _{V\oplus W}}$ is given as

$${\displaystyle g\mapsto {\begin{pmatrix}\rho _{V}(g)&0\\0&\rho _{W}(g)\end{pmatrix}}.}$$

Moreover, if we treat ${\displaystyle V}$ and ${\displaystyle W}$ as modules over the group ring ${\displaystyle kG}$, where ${\displaystyle k}$ is the field, then the direct sum of the representations ${\displaystyle V}$ and ${\displaystyle W}$ is equal to their direct sum as ${\displaystyle kG}$ modules.

Some authors will speak of the direct sum ${\displaystyle R\oplus S}$ of two rings when they mean the direct product ${\displaystyle R\times S}$, but this should be avoided[3] since ${\displaystyle R\times S}$ does not receive natural ring homomorphisms from R and S: in particular, the map ${\displaystyle R\to R\times S}$ sending r to (r,0) is not a ring homomorphism since it fails to send 1 to (1,1) (assuming that 0≠1 in S). Thus ${\displaystyle R\times S}$ is not a coproduct in the category of rings, and should not be written as a direct sum. (The coproduct in the category of commutative rings is the tensor product of rings.[4] In the category of rings, the coproduct is given by a construction similar to the free product of groups.)

Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If ${\displaystyle (R_{i})_{i\in I}}$ is an infinite collection of nontrivial rings, then the direct sum of the underlying additive groups can be equipped with termwise multiplication, but this produces a rng, i.e., a ring without a multiplicative identity.

An additive category is an abstraction of the properties of the category of modules.[5][6] In such a category finite products and coproducts agree and the direct sum is either of them, cf. biproduct.

General case:[7] In category theory the direct sum is often, but not always, the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules.