dual module

Let $R$ be a ring and $M$ be a left http://planetmath.org/node/365 $R$ -module. The dual module of $M$ isthe right http://planetmath.org/node/365 $R$ -module consisting of all module homomorphisms from $M$ into $R$ .

It is denoted by $M^{\\ast}$ . The elements of $M^{\\ast}$ are called linear functionals.

The action of $R$ on $M^{\\ast}$ is given by $(fr)(m)=(f(m))r$ for $f\\in M^{\\ast}$ , $m\\in M$ , and $r\\in R$ .

If $R$ is commutative, then every $M$ is an http://planetmath.org/node/987 $(R,R)$ -bimodule with $rm=mr$ for all $r\\in R$ and $m\\in M$ . Hence, it makes sense to ask whether $M$ and $M^{\\ast}$ are isomorphic. Suppose that $b:M\\times M\\to R$ is a bilinear form. Then it is easy to check that for a fixed $m\\in M$ , the function $b(m,-):M\\to R$ is a module homomorphism,so is an element of $M^{\\ast}$ . Then we have a module homomorphism from $M$ to $M^{\\ast}$ given by $m\\mapsto b(m,-)$ .

单词	DualModule
释义	dual module Let $R$ be a ring and $M$ be a left http://planetmath.org/node/365 $R$ -module. The dual module of $M$ isthe right http://planetmath.org/node/365 $R$ -module consisting of all module homomorphisms from $M$ into $R$ . It is denoted by $M^{\\ast}$ . The elements of $M^{\\ast}$ are called linear functionals. The action of $R$ on $M^{\\ast}$ is given by $(fr)(m)=(f(m))r$ for $f\\in M^{\\ast}$ , $m\\in M$ , and $r\\in R$ . If $R$ is commutative, then every $M$ is an http://planetmath.org/node/987 $(R,R)$ -bimodule with $rm=mr$ for all $r\\in R$ and $m\\in M$ . Hence, it makes sense to ask whether $M$ and $M^{\\ast}$ are isomorphic. Suppose that $b:M\\times M\\to R$ is a bilinear form. Then it is easy to check that for a fixed $m\\in M$ , the function $b(m,-):M\\to R$ is a module homomorphism,so is an element of $M^{\\ast}$ . Then we have a module homomorphism from $M$ to $M^{\\ast}$ given by $m\\mapsto b(m,-)$ .
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