The direct sum is an operation from abstract algebra, a branch of mathematics. For example, the direct sum
Contents
- Examples
- Internal and external direct sums
- Direct sum of abelian groups
- Direct sum of modules
- Direct sum of group representations
- Direct sum of rings
- Direct sum in categories
- Homomorphisms
- References
We can also form direct sums with any number of summands, for example
In the case of two summands, or any finite number of summands, the direct sum is the same as the direct product. If the arithmetic operation is written as +, as it usually is in abelian groups, then we use the direct sum. If the arithmetic operation is written as × or ⋅ or using juxtaposition (as in the expression
In the case where infinitely many objects are combined, most authors make a distinction between direct sum and direct product. As an example, consider the direct sum and direct product of infinitely many real lines. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there would be a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. More generally, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are
Examples
For example, the xy-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces, namely the x and y axes. In this direct sum, the x and y axes intersect only at the origin (the zero vector). Addition is defined coordinate-wise, that is
Given two objects
Internal and external direct sums
A distinction is made between internal and external direct sums, though the two are isomorphic. If the factors are defined first, and then the direct sum is defined in terms of the factors, we have an external direct sum. For example, if we define the real numbers
If, on the other hand, we first define some algebraic object,
Direct sum of abelian groups
The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups
This definition generalizes to direct sums of finitely many abelian groups.
For an infinite family of abelian groups Ai for i ∈ I, the direct sum
is a proper subgroup of the direct product. It consists of the elements
Direct sum of modules
The direct sum of modules is a construction which combines several modules into a new module.
The most familiar examples of this construction occur when considering vector spaces, which are modules over a field. The construction may also be extended to Banach spaces and Hilbert spaces.
Direct sum of group representations
The direct sum of group representations generalizes the direct sum of the underlying modules, adding a group action to it. Specifically, given a group G and two representations V and W of G (or, more generally, two G-modules), the direct sum of the representations is V ⊕ W with the action of g ∈ G given component-wise, i.e.
g·(v, w) = (g·v, g·w).Direct sum of rings
Some authors will speak of the direct sum
Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If
Direct sum in categories
An additive category is an abstraction of the properties of the category of modules.
In such a category finite products and coproducts agree and the direct sum is either of them, cf. biproduct.
General case : In category theory the direct sum is often, but not always, the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules.
Homomorphisms
The direct sum