direct sum

Let {Xi:iI} be a collectionMathworldPlanetmath of modules in some categoryMathworldPlanetmath of modules. Then the direct sumMathworldPlanetmathPlanetmathPlanetmath iIXi of that collection is the submoduleMathworldPlanetmath of the direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( of the Xi consisting of all elements (xi) such that all but a finite number of the xi are zero.

For each jI we have a projection pj:iIXiXj defined by (xi)xj, and an injection λj:XjiIXi where an element xj of Xj maps to the element of iIXi whose jth term is xj and every other term is zero.

The direct sum iIXi satisfies a certain universal propertyMathworldPlanetmath. Namely, if Y is a module and there exist homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath fi:YXi for all iI, then there exists a unique homomorphism ϕ:iIXiY satisfying piϕ=fi for all iI.


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 iIXi.

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