direct product of modules
Let be a collection of modules in some category of modules. Then the direct product of that collection is the module whose underlying set is the Cartesian product of the with componentwise addition and scalar multiplication. For example, in a category of left modules:
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 product 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 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 |
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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 |