direct sum
Let be a collection of modules in some category of modules. Then the direct sum of that collection is the submodule of the direct product (http://planetmath.org/DirectProduct) of the consisting of all elements such that all but a finite number of the are zero.
For each we have a projection defined by , and an injection where an element of maps to the element of whose th term is and every other term is zero.
The direct sum satisfies a certain universal property. Namely, if is a module and there exist homomorphisms for all , then there exists a unique homomorphism satisfying for all .
The direct sum is often referred to as the weak direct sum or simply the sum.
Compare this to the direct product of modules.
Often an internal direct sum is written as .
Title | direct sum |
---|---|
Canonical name | DirectSum |
Date of creation | 2013-03-22 12:09:37 |
Last modified on | 2013-03-22 12:09:37 |
Owner | antizeus (11) |
Last modified by | antizeus (11) |
Numerical id | 10 |
Author | antizeus (11) |
Entry type | Definition |
Classification | msc 16-00 |
Synonym | weak direct sum |
Related topic | CategoricalDirectSum |
Related topic | DirectSummand |
Related topic | DirectSumOfMatrices |