Direct sum

The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. As an example, the direct sum of two abelian groups $$A$$ and $$B$$ is another abelian group $$A\oplus B$$ consisting of the ordered pairs $$(a,b)$$ where $$a \in A$$ and $$b \in B$$. To add ordered pairs, we define the sum $$(a, b) + (c, d)$$ to be $$(a + c, b + d)$$; in other words addition is defined coordinate-wise. For example, the direct sum $$ \Reals \oplus \Reals $$, where $$ \Reals $$ is real coordinate space, is the Cartesian plane, $$ \R ^2 $$. A similar process can be used to form the direct sum of two vector spaces or two modules.

We can also form direct sums with any finite number of summands, for example $$A \oplus B \oplus C$$, provided $$A, B,$$ and $$C$$ are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). This relies on the fact that the direct sum is associative up to isomorphism. That is, $$(A \oplus B) \oplus C \cong A \oplus (B \oplus C)$$ for any algebraic structures $$A$$, $$B$$, and $$C$$ of the same kind. The direct sum is also commutative up to isomorphism, i.e. $$A \oplus B \cong B \oplus A$$ for any algebraic structures $$A$$ and $$B$$ of the same kind.

The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups.

In the case where infinitely many objects are combined, the direct sum and direct product are not isomorphic, even for abelian groups, vector spaces, or modules. As an example, consider the direct sum and direct product of (countably) infinitely many copies of the integers. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there is a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. Often, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are $$(A_i)_{i \in I}$$, the direct sum $$\bigoplus_{i \in I} A_i$$ is defined to be the set of tuples $$(a_i)_{i \in I}$$ with $$a_i \in A_i$$ such that $$a_i=0$$ for all but finitely many i. The direct sum $\bigoplus_{i \in I} A_i$ is contained in the direct product $\prod_{i \in I} A_i$, but is strictly smaller when the index set $$I$$ is infinite, because an element of the direct product can have infinitely many nonzero coordinates.

Examples
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). Addition is defined coordinate-wise, that is $$(x_1,y_1) + (x_2,y_2) = (x_1+x_2, y_1 + y_2)$$, which is the same as vector addition.

Given two structures $$A$$ and $$B$$, their direct sum is written as $$A\oplus B$$. Given an indexed family of structures $$A_i$$, indexed with $$i \in I$$, the direct sum may be written $ A=\bigoplus_{i\in I}A_i$. Each Ai is called a direct summand of A. If the index set is finite, the direct sum is the same as the direct product. In the case of groups, if the group operation is written as $$+$$ the phrase "direct sum" is used, while if the group operation is written $$*$$ the phrase "direct product" is used. When the index set is infinite, the direct sum is not the same as the direct product since the direct sum has the extra requirement that all but finitely many coordinates must be zero.

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

If, on the other hand, we first define some algebraic structure $$S$$ and then write $$S$$ as a direct sum of two substructures $$V$$ and $$W$$, then the direct sum is said to be internal. In this case, each element of $$S$$ is expressible uniquely as an algebraic combination of an element of $$V$$ and an element of $$W$$. For an example of an internal direct sum, consider $$\mathbb Z_6$$ (the integers modulo six), whose elements are $$\{0, 1, 2, 3, 4, 5\}$$. This is expressible as an internal direct sum $$\mathbb Z_6 = \{0, 2, 4\} \oplus \{0, 3\}$$.

Direct sum of abelian groups
The direct sum of abelian groups is a prototypical example of a direct sum. Given two such groups $$(A, \circ)$$ and $$(B, \bullet),$$ their direct sum $$A \oplus B$$ is the same as their direct product. That is, the underlying set is the Cartesian product $$A \times B$$ and the group operation $$\,\cdot\,$$ is defined component-wise: $$\left(a_1, b_1\right) \cdot \left(a_2, b_2\right) = \left(a_1 \circ a_2, b_1 \bullet b_2\right).$$ This definition generalizes to direct sums of finitely many abelian groups.

For an arbitrary family of groups $$A_i$$ indexed by $$i \in I,$$ their $$\bigoplus_{i \in I} A_i$$ is the subgroup of the direct product that consists of the elements $\left(a_i\right)_{i \in I} \in \prod_{i \in I} A_i$ that have finite support, where by definition, $$\left(a_i\right)_{i \in I}$$ is said to have  if $$a_i$$ is the identity element of $$A_i$$ for all but finitely many $$i.$$ The direct sum of an infinite family $$\left(A_i\right)_{i \in I}$$ of non-trivial groups is a proper subgroup of the product group $\prod_{i \in I} A_i.$

Direct sum of modules
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.

Direct sum in categories
An additive category is an abstraction of the properties of the category of modules. In such a category, finite products and coproducts agree and the direct sum is either of them, cf. biproduct.

General case: In category theory the 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.

Direct sums versus coproducts in category of groups
However, the direct sum $$S_3 \oplus \Z_2$$ (defined identically to the direct sum of abelian groups) is a coproduct of the groups $$S_3$$ and $$\Z_2$$ in the category of groups. So for this category, a categorical direct sum is often simply called a coproduct to avoid any possible confusion.

Direct sum of group representations
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 $$V \oplus W$$ with the action of $$g \in G$$ given component-wise, that is, $$g \cdot (v, w) = (g \cdot v, g \cdot w).$$ Another equivalent way of defining the direct sum is as follows:

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

Furthermore, if $$V,\,W$$ are finite dimensional, then, given a basis of $$V,\,W$$, $$\rho_V$$ and $$\rho_W$$ are matrix-valued. In this case, $$\rho_{V \oplus W}$$ is given as $$g \mapsto \begin{pmatrix}\rho_V(g) & 0 \\ 0 & \rho_W(g)\end{pmatrix}.$$

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

Direct sum of rings
Some authors will speak of the direct sum $$R \oplus S$$ of two rings when they mean the direct product $$R \times S$$, but this should be avoided since $$R \times S$$ does not receive natural ring homomorphisms from $$R$$ and $$S$$: in particular, the map $$R \to R \times S$$ sending $$r$$ to $$(r, 0)$$ is not a ring homomorphism since it fails to send 1 to $$(1, 1)$$ (assuming that $$0 \neq 1$$ in $$S$$). Thus $$R \times S$$ 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. 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 $$(R_i)_{i \in I}$$ 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, that is, a ring without a multiplicative identity.

Direct sum of matrices
For any arbitrary matrices $$\mathbf{A}$$ and $$\mathbf{B}$$, the direct sum $$\mathbf{A} \oplus \mathbf{B}$$ is defined as the block diagonal matrix of $$\mathbf{A}$$ and $$\mathbf{B}$$ if both are square matrices (and to an analogous block matrix, if not). $$\mathbf{A} \oplus \mathbf{B} = \begin{bmatrix} \mathbf{A} & 0         \\ 0         & \mathbf{B} \end{bmatrix}.$$

Direct sum of topological vector spaces
A topological vector space (TVS) $$X,$$ such as a Banach space, is said to be a of two vector subspaces $$M$$ and $$N$$ if the addition map $$\begin{alignat}{4} \ \;&& M \times N &&\;\to    \;& X \\[0.3ex] && (m, n) &&\;\mapsto\;& m + n \\ \end{alignat}$$ is an isomorphism of topological vector spaces (meaning that this linear map is a bijective homeomorphism), in which case $$M$$ and $$N$$ are said to be in $$X.$$ This is true if and only if when considered as additive topological groups (so scalar multiplication is ignored), $$X$$ is the topological direct sum of the topological subgroups $$M$$ and $$N.$$ If this is the case and if $$X$$ is Hausdorff then $$M$$ and $$N$$ are necessarily closed subspaces of $$X.$$

If $$M$$ is a vector subspace of a real or complex vector space $$X$$ then there always exists another vector subspace $$N$$ of $$X,$$ called an such that $$X$$ is the  of $$M$$ and $$N$$ (which happens if and only if the addition map $$M \times N \to X$$ is a vector space isomorphism). In contrast to algebraic direct sums, the existence of such a complement is no longer guaranteed for topological direct sums.

A vector subspace $$M$$ of $$X$$ is said to be a  if there exists some vector subspace $$N$$ of $$X$$ such that $$X$$ is the topological direct sum of $$M$$ and $$N.$$ A vector subspace is called if it is not a complemented subspace. For example, every vector subspace of a Hausdorff TVS that is not a closed subset is necessarily uncomplemented. Every closed vector subspace of a Hilbert space is complemented. But every Banach space that is not a Hilbert space necessarily possess some uncomplemented closed vector subspace.

Homomorphisms
The direct sum $\bigoplus_{i \in I} A_i$ comes equipped with a projection homomorphism $\pi_j \colon \, \bigoplus_{i \in I} A_i \to A_j$  for each j in I and a coprojection $\alpha_j \colon \, A_j \to \bigoplus_{i \in I} A_i$  for each j in I. Given another algebraic structure $$B$$ (with the same additional structure) and homomorphisms $$g_j \colon A_j \to B$$ for every j in I, there is a unique homomorphism $g \colon \, \bigoplus_{i \in I} A_i \to B$, called the sum of the gj, such that $$g \alpha_j =g_j$$ for all j. Thus the direct sum is the coproduct in the appropriate category.