dual module
Let be a ring and be a left http://planetmath.org/node/365-module. The dual module of is the right http://planetmath.org/node/365-module consisting of all module homomorphisms from into .
It is denoted by . The elements of are called linear functionals.
The action of on is given by for , , and .
If is commutative, then every is an http://planetmath.org/node/987-bimodule with for all and . Hence, it makes sense to ask whether and are isomorphic. Suppose that is a bilinear form. Then it is easy to check that for a fixed , the function is a module homomorphism, so is an element of . Then we have a module homomorphism from to given by .
Title | dual module |
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Canonical name | DualModule |
Date of creation | 2013-03-22 16:00:26 |
Last modified on | 2013-03-22 16:00:26 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 10 |
Author | Mathprof (13753) |
Entry type | Definition |
Classification | msc 16-00 |
Related topic | Unimodular |
Defines | linear functional |