Direct Sum - Internal Direct Sum

An internal direct sum is simply a direct sum of subobjects of an object.

For example, the real vector space R2 = {(x, y) : x, y ∈ R} is the direct sum of the x-axis {(x, 0) : x ∈ R} and the y-axis {(0, y) : y ∈ R}, and the sum of (x, 0) and (0, y) is the "internal" sum in the vector space R2; thus, this is an internal direct sum. More generally, given a vector space V and two subspaces U and W, V is the (internal) direct sum U ⊕ W if

1. U + W = {u + w : u ∈ U, w ∈ W} = V, and
2. if u + w = 0 with u ∈ U and w ∈ W, then u = w = 0.

In other words, every element of V can be written uniquely as the sum of an element in U with an element of W

Another case is that of abelian groups. For example, the Klein four-group V = {e, a, b, ab} is the (internal) direct sum of the cyclic subgroups <a> and <b>.

By contrast, a direct sum of two objects which are not subobjects of a common object is an external direct sum. Note however that "external direct sum" is also used to refer to an infinite direct sum of groups, to contrast with the (larger) direct product.