We give the construction first in these two cases, under the assumption that we have only two objects.
The key elements of the general construction are more clearly identified by considering these two cases in depth.
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 (Halmos 1974, §18) by defining the operations componentwise: for v, v1, v2 ∈ V, w, w1, w2 ∈ 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:
This construction readily generalizes to any finite number of vector spaces.
Thus the Cartesian product G × H is equipped with the structure of an abelian group by defining the operations componentwise: for g1, g2 in G, and h1, h2 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:
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.
Let R be a ring, and {Mi : i ∈ I} a family of left R-modules indexed by the set I.
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
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
In this case, M is naturally isomorphic to the (external) direct sum of the Mi as defined above (Adamson 1972, p.61).
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 Mi to those functions which are zero for all arguments but i.
In fact, subtraction can be defined, and every commutative monoid can be extended to an abelian group.
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.
If the modules we are considering carry some additional structure (for example, 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.
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.
while for the direct product a scalar factor may be collected alternately with the parts, but not both:
as rings of scalars in his analysis of Clifford Algebras and the Classical Groups (1995).
in which the summation makes sense even for infinite index sets
are given, one can construct their orthogonal direct sum as above (since they are vector spaces), defining the inner product as:
are given, we can carry out the same construction; notice that when defining the inner product, only finitely many summands will be non-zero.
Alternatively and equivalently, one can define the direct sum of the Hilbert spaces
Every Hilbert space is isomorphic to a direct sum of sufficiently many copies of the base field, which is either
This is equivalent to the assertion that every Hilbert space has an orthonormal basis.