direct product of modules

Let {Xi:iI} be a collectionMathworldPlanetmath of modules in some categoryMathworldPlanetmath of modules. Then the direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath iIXi of that collection is the module whose underlying set is the Cartesian productMathworldPlanetmath of the Xi with componentwise addition and scalar multiplication. For example, in a category of left modules:


For each jI we have a projectionPlanetmathPlanetmath pj:iIXiXj defined by (xi)xj, and an injectionMathworldPlanetmath 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 product iIXi satisfies a certain universal propertyMathworldPlanetmath. Namely, if Y is a module and there exist homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath fi:XiY for all iI, then there exists a unique homomorphism ϕ:YiIXi satisfying ϕλi=fi for all iI.


The direct product is often referred to as the complete direct sum, or the strong direct sum, or simply the .

Compare this to the direct sum of modules.

Title direct product of modules
Canonical name DirectProductOfModules
Date of creation 2013-03-22 12:09:34
Last modified on 2013-03-22 12:09:34
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 10
Author Mathprof (13753)
Entry type Definition
Classification msc 16D10
Synonym strong direct sum
Synonym complete direct sum
Related topic CategoricalDirectProduct
Defines direct product