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 |