Dual Modules

Let R be a commutative ring. For two (left) R-modules M and N , the set HomR(M,N) of allR-linear maps fromM toN is anR-module under natural addition and scaling operations on linear maps. (If R were non-commutative then the definition (r · f)(m) = r · (f(m)) would yield a function r · f from M to N which is usually not R-linear. Try it!) In the special case where N = R we get the R-module M∨ = HomR(M,R). This is called the dual module, dual space, or R-dual of M . Elements of M∨ are called linear functionals or simply functionals. Here are some places in mathematics where dual modules show up: (1) linear algebra: coordinate functions on Rn. (2) analysis: integration on a space of continuous functions. (3) geometry: the abstract definition of a tangent space (directional derivatives). (4) number theory: the different ideal of a number field. There really is no picture of the dual module, but its elements could be thought of as “potential coordinate functions” on M (plus the function 0, so we have a module). This idea is accurate if M if a finite-dimensional vector space, or even a finite-free module, but in more general settings it can be an oversimplification. In Section 2 we will look at some examples of dual modules. The behavior of the dual module on direct sums and direct products is the topic of Section 3. The special case of dual modules for finite free modules is in Section 4, where we meet the important double duality isomorphism. Section 5 describes the construction of the dual of a linear map between modules, which generalizes the matrix transpose. In Section 6 we will see how dual modules arise in concrete ways using the language of (perfect) pairings.