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 |
---|---|
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 |