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.

See the article decomposition of a module for a way to write a module as a direct sum of submodules.

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.

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:

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.

This construction readily generalizes to any finite number of vector spaces.

For abelian groups G and H which are written additively, the direct product of G and H is also called a direct sum (Mac Lane & Birkhoff 1999, §V.6). Thus the Cartesian product G × H is equipped with the structure of an abelian group by defining the operations componentwise:

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.

This construction readily generalises to any finite number of abelian groups.

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 (Bourbaki 1989, §II.1.6).

Suppose M is some R-module, and Mi 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 Mi, then we say that M is the internal direct sum of the submodules Mi (Halmos 1974, §18). In this case, M is naturally isomorphic to the (external) direct sum of the Mi as defined above (Adamson 1972, p.61).

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.

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. If fi : MiM are arbitrary R-linear maps for every i, then there exists precisely one R-linear map

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.

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. Two prominent examples occur for Banach spaces and Hilbert spaces.

In some classical texts, the phrase "direct sum of algebras over a field" is also introduced for denoting the algebraic structure that is presently more commonly called a direct product of algebras; that is, the Cartesian product of the underlying sets with the componentwise operations. 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).

The construction described above, as well as Wedderburn's use of the terms direct sum and direct product follow a different convention than 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.

The norm is given by the sum above. The direct sum with this norm is again a Banach space.

The resulting direct sum is a Hilbert space which contains the given Hilbert spaces as mutually orthogonal subspaces.