Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces.
Construction for vector spaces and abelian groups.
We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by considering these two cases in depth.
Construction for two vector spaces.
Suppose "V" and "W" are vector spaces over the field "K". The cartesian product "V" × "W" can be given the structure of a vector space over "K" by defining the operations componentwise:
for "v", "v"1, "v"2 ∈ "V", "w", "w"1, "w"2 ∈ "W", and "α" ∈ "K".
The resulting vector space is called the "direct sum" of "V" and "W" and is usually denoted by a plus symbol inside a circle:
It is customary to write the elements of an ordered sum not as ordered pairs ("v", "w"), but as a sum "v" + "w".
The subspace "V" × {0} of "V" ⊕ "W" is isomorphic to "V" and is often identified with "V"; similarly for {0} × "W" and "W". (See "internal direct sum" below.) With this identification, every element of "V" ⊕ "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" ⊕ "W" is equal to the sum of the dimensions of "V" and "W". One elementary use is the reconstruction
of a finite vector space from any subspace "W" and its orthogonal complement: 
This construction readily generalizes to any finite number of vector spaces.
Construction for two abelian groups.
For abelian groups "G" and "H" which are written additively, the direct product of "G" and "H" is also called a direct sum . Thus the Cartesian product "G" × "H" is equipped with the structure of an abelian group by defining the operations componentwise:
for "g"1, "g"2 in "G", and "h"1, "h"2 in "H".
Integral multiples are similarly defined componentwise by
for "g" in "G", "h" in "H", and "n" an integer. This parallels the extension of the scalar product of vector spaces to the direct sum above.
The resulting abelian group is called the "direct sum" of "G" and "H" and is usually denoted by a plus symbol inside a circle:
It is customary to write the elements of an ordered sum not as ordered pairs ("g", "h"), but as a sum "g" + "h".
The subgroup "G" × {0} of "G" ⊕ "H" is isomorphic to "G" and is often identified with "G"; similarly for {0} × "H" and "H". (See "internal direct sum" below.) With this identification, it is true that every element of "G" ⊕ "H" can be written in one and only one way as the sum of an element of "G" and an element of "H". The rank of "G" ⊕ "H" is equal to the sum of the ranks of "G" and "H".
This construction readily generalises to any finite number of abelian groups.
Construction for an arbitrary family of modules.
One should notice a clear similarity between the definitions of the direct sum of two vector spaces and of two abelian groups. In fact, each is a special case of the construction of the direct sum of two modules. Additionally, by modifying the definition one can accommodate the direct sum of an infinite family of modules. The precise definition is as follows .
Let "R" be a ring, and {"M""i" : "i" ∈ "I"} a family of left "R"-modules indexed by the set "I". The "direct sum" of {"M""i"} is then defined to be the set of all sequences formula_4 where formula_5 and formula_6 for cofinitely many indices "i". (The direct product is analogous but the indices do not need to cofinitely vanish.)
It can also be defined as functions α from "I" to the disjoint union of the modules "M""i" such that α("i") ∈ "M""i" for all "i" ∈ "I" and α("i") = 0 for cofinitely many indices "i". These functions can equivalently be regarded as finitely supported sections of the fiber bundle over the index set "I", with the fiber over formula_7 being formula_8.
This set inherits the module structure via component-wise addition and scalar multiplication. Explicitly, two such sequences (or functions) α and β can be added by writing formula_9 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 defining formula_10 for all "i". In this way, the direct sum becomes a left "R"-module, and it is denoted
It is customary to write the sequence formula_4 as a sum formula_13. Sometimes a primed summation formula_14 is used to indicate that cofinitely many of the terms are zero.
Internal direct sum.
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 .
A submodule "N" of "M" is a direct summand of "M" if there exists some other submodule "N′" of "M" such that "M" is the "internal" direct sum of "N" and "N′". In this case, "N" and "N′" are complementary submodules.
Universal property.
In the language of category theory, the direct sum is a coproduct and hence a colimit 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"
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
such that "f" o "ji" = "f""i" for all "i".
Grothendieck group.
The direct sum gives a collection of objects the structure of a commutative monoid, in that the addition of objects is defined, but not subtraction. In fact, subtraction can be defined, and every commutative monoid can be extended to an abelian group. This extension is known as the Grothendieck group. The extension is done by defining equivalence classes of pairs of objects, which allows certain pairs to be treated as inverses. The construction, detailed in the article on the Grothendieck group, is "universal", in that it has the universal property of being unique, and homomorphic to any other embedding of a commutative monoid in an abelian group.
Direct sum of modules with additional structure.
If the modules we are considering carry some additional structure (e.g. a norm or an inner product), then the direct sum of the modules can often be made to carry this additional structure, as well. In this case, we obtain the coproduct in the appropriate category of all objects carrying the additional structure. Two prominent examples occur for Banach spaces and Hilbert spaces.
In some classical texts, the notion of direct sum of algebras over a field is also introduced. This construction, however, does not provide a coproduct in the category of algebras, but a direct product ("see note below" and the remark on direct sums of rings).
Direct sum of algebras.
A direct sum of algebras "X" and "Y" is the direct sum as vector spaces, with product 
Consider these classical examples:
Joseph Wedderburn exploited the concept of a direct sum of algebras in his classification of hypercomplex numbers. See his "Lectures on Matrices" (1934), page 151.
Wedderburn makes clear the distinction between a direct sum and a direct product of algebras: For the direct sum the field of scalars acts jointly on both parts: formula_27 while for the direct product a scalar factor may be collected alternately with the parts, but not both: formula_28.
Ian R. Porteous uses the three direct sums above, denoting them formula_29, as rings of scalars in his analysis of "Clifford Algebras and the Classical Groups" (1995).
The construction described above, as well as Wedderburn's use of the terms "direct sum" and "direct product" follow a different convention from the one in category theory. In categorical terms, Wedderburn's "direct sum" is a categorical product, whilst Wedderburn's "direct product" is a coproduct (or categorical sum), which (for commutative algebras) actually corresponds to the tensor product of algebras.
Composition algebras.
A composition algebra ("A", *, "n") is an algebra over a field "A", an involution * and a "norm" "n"("x") = "x x"*. Any field "K" gives rise to a series of composition algebras beginning with "K", and the trivial involution, so that "n"("x") = "x"2. The inductive step in the series involves forming the direct sum "A" ⊕ "A" and using the new involution formula_30
Leonard Dickson developed this construction doubling quaternions for Cayley numbers, and the doubling method involving the direct sum "A" ⊕ "A" is called the Cayley–Dickson construction. In the instance beginning with "K" = ℝ, the series generates complex numbers, quaternions, octonions, and sedenions. Beginning with "K" = ℂ and the norm "n"("z") = "z"2, the series continues with bicomplex numbers, biquaternions, and bioctonions.
Max Zorn realized that the classical Cayley–Dickson construction missed constructing some composition algebras that arise as real subalgebras in the (ℂ, "z"2) series, in particular the split-octonions. A modified Cayley–Dickson construction, still based on use of the direct sum "A" ⊕ "A" of a base algebra "A", has since been used to exhibit the series ℝ, split-complex numbers, split-quaternions, and split-octonions.
Direct sum of Banach spaces.
The direct sum of two Banach spaces "X" and "Y" is the direct sum of "X" and "Y" considered as vector spaces, with the norm ||("x","y")|| = ||"x"||X + ||"y"||Y for all "x" in "X" and "y" in "Y".
Generally, if "X""i" is a collection of Banach spaces, where "i" traverses the index set "I", then the direct sum ⨁"i"∈"I" "X""i" is a module consisting of all functions "x" defined over "I" such that "x"("i") ∈ "X""i" for all "i" ∈ "I" and
The norm is given by the sum above. The direct sum with this norm is again a Banach space.
For example, if we take the index set "I" = N and "X""i" = R, then the direct sum ⨁"i"∈N"X""i" is the space "l"1, which consists of all the sequences ("a""i") of reals with finite norm ||"a"|| = ∑"i" |"a""i"|.
A closed subspace "A" of a Banach space "X" is complemented if there is another closed subspace "B" of "X" such that "X" is equal to the internal direct sum formula_32. Note that not every closed subspace is complemented, e.g. "c"0 is not complemented in formula_33.
Direct sum of modules with bilinear forms.
Let {("M""i","b""i")  : "i" ∈ "I"} be a family indexed by "I" of modules equipped with bilinear forms. The orthogonal direct sum is the module direct sum with bilinear form "B" defined by
in which the summation makes sense even for infinite index sets "I" because only finitely many of the terms are non-zero.
Direct sum of Hilbert spaces.
If finitely many Hilbert spaces "H"1...,"H""n" are given, one can construct their orthogonal direct sum as above (since they are vector spaces), defining the inner product as:
The resulting direct sum is a Hilbert space which contains the given Hilbert spaces as mutually orthogonal subspaces.
If infinitely many Hilbert spaces "H""i" for "i" in "I" are given, we can carry out the same construction; notice that when defining the inner product, only finitely many summands will be non-zero. However, the result will only be an inner product space and it will not necessarily be complete. We then define the direct sum of the Hilbert spaces "H""i" to be the completion of this inner product space. 
Alternatively and equivalently, one can define the direct sum of the Hilbert spaces "H""i" as the space of all functions α with domain "I", such that α("i") is an element of "H""i" for every "i" in "I" and:
The inner product of two such function α and β is then defined as:
This space is complete and we get a Hilbert space.
For example, if we take the index set "I" = N and "X""i" = R, then the direct sum ⨁"i"∈N "X""i" is the space "l"2, which consists of all the sequences ("a""i") of reals with finite norm formula_38. Comparing this with the example for Banach spaces, we see that the Banach space direct sum and the Hilbert space direct sum are not necessarily the same. But if there are only finitely many summands, then the Banach space direct sum is isomorphic to the Hilbert space direct sum, although the norm will be different.
Every Hilbert space is isomorphic to a direct sum of sufficiently many copies of the base field (either R or C). This is equivalent to the assertion that every Hilbert space has an orthonormal basis. More generally, every closed subspace of a Hilbert space is complemented: it admits an orthogonal complement. Conversely, the Lindenstrauss–Tzafriri theorem asserts that if every closed subspace of a Banach space is complemented, then the Banach space is isomorphic (topologically) to a Hilbert space.