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.

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 (Halmos 1974, §18) by defining the operations componentwise:

(v1, w1) + (v2, w2) = (v1 + v2, w1 + w2)

α (v, w) = (α v, α w)

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:

V

⊕

W

{\displaystyle V\oplus W}

It is customary to write the elements of an ordered sum not as ordered pairs (v, w), but as a sum v + w.